For 2x2 Loopover, God's number is 4. There are only 24 possible scrambles so it is possible to brute-force the entire configuration space. A 4-move scramble is:
DC
BA

Disclaimer: No new results.
Slightly tighter counting of 5x5 move sequences at fixed depths, continuing from xyzzy's post #6 (https://www.speedsolving.com/threads/loopover-gods-number-upper-bounds-4×4-asymptotics-etc.75180/#post-1333877).
On a 5x5, the move sequences R1 C0 R0 C0' and C0 R0 C0'...

5x5 NRG GN is at most 278.
I used the solving phases null --> LMN QRS --> LMNO QRST VWXY --> C HIJ LMNO QRST VWXY --> entire board. 25+65+82+106=278.
(Sidenote: for 4x4NRG, using the solving phases null --> IJKL MNOP --> entire board yields a bound of 28+94=122, which is worse than xyzzy's...

5x5 GN is at most 46.
I reduced the GN of 5x5: 0x0 --> 3x3 to 18, using several new techniques:
First, I split this solving stage into the phases null --> AC KM --> 3x3 (ABC FGH KLM) instead of null --> AB FG --> 3x3, which decreases the scrambles I would have to process from ~233e9 to ~188e9...

3-cycles and 2,2-cycles + Dijkstra's algorithm
This is similar to doing a standard Loopover BFS, except the transitions are 3-cycles and 2,2-cycles instead of single moves, and we have to use Dijkstra's algorithm instead of BFS (since we are now evaluating shortest paths in a weighted graph)...

A much better upper bound technique:
This one comes partially from xyzzy via a Discord post. As in regular Loopover, we can solve a NRG board by extending a block of solved pieces. Unlike in regular Loopover, when extending an existing block to a bigger block, the initial block must leave at...

New upper bound technique:
Let C3(a,b,c) be the minimum number of moves needed to cycle locations a-->b-->c-->a and C4(a,b,c,d) be the minimum number of moves needed to swap locations a<-->b and c<-->d.
Consider for simplicity only using the 3-cycle. Suppose the piece at the location (0,0) is...

New upper bounds:
I added permutation "symmetries" to the 3-cycle and double-swap algorithms. Code is at https://github.com/coolcomputery/Loopover-NRG-Upper-Bounds.
Basically, every algorithm A describes an action (L,P) for some array L of locations on the NxN board and a permutation P, where...

5x5 GN is at most 49.
I reduced the GN for 0x0->2x2->3x3 (i.e. solving AB FG, then C H KLM) to at most 21 moves., by considering every scramble of the pieces ABC FGH KLM that would take >21 moves by naively applying solving the two phases, applying at most 2 prefix moves, and solving the two...

6x6 GN is at most 101.
I reduced the GN of 4x5->5x5 + 5x5->6x6 to 38, by applying at most 5 prefix moves to every scramble that would naively take >38 moves. There were 249347110 total scrambles, which took 4294.610 seconds to process (including making the two BFS trees). Referring to this...

Oops, I had F gripped instead of A (it's the default gripped piece on https://loopover.xyz/). Here's the scramble again, this time with A gripped:
ABCD FBCJ
EFGH --> EPGL
IJKL NMIA
MNOP KHDO

New lower bounds:
Using a similar technique by xyzzy for regular Loopover, we can describe any NRG scramble as a list of "syllables", where a syllable is a run of Ds, Rs, Ls, or Us, and syllables alternate between horizontal (Rs and Ls) and vertical (Ds and Us). Then, for a NxN board, let S be...

Loopover NRG is like Loopover, but you can only "grip" a certain fixed piece at all times, i.e. you can only slide the row or the column that that piece is in. Here is an example solve (not by me).
Although there's been research on God's number for regular Loopover, I haven't found any for...

4x4 STM GD is at most 22
I added row reflection, column reflection, and transposition (flipping along the main diagonal) to my previous algorithm on improving the 0x0->2x3 and 2x3->4x4 block-building.
A total of 1647318090782 scrambles that required >=23 moves when the block-building was applied...

5x5 STM GD is indeed at most 53
I improved the 0x0->2x3 and the 2x3->3x4 steps differently: instead of solving translated regions of each scramble (which requires testing over all permutations of where the non-3x4 pieces can go in a 5x5), I added 1 prefix move before running the block-building...

4x4 STM GD is at most 23
https://github.com/coolcomputery/Loopover-Brute-Force-and-Improvement/blob/master/LoopoverBruteForce.java
The solved 4x4 board will look like this:
0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
1. Enumerate all ways of building the 2x3 whose upper-left corner is 0, 0...

For 2x2 Loopover, God's number is 4. There are only 24 possible scrambles so it is possible to brute-force the entire configuration space. A 4-move scramble is:
DC
BA

Disclaimer: No new results.
Slightly tighter counting of 5x5 move sequences at fixed depths, continuing from xyzzy's post #6 (https://www.speedsolving.com/threads/loopover-gods-number-upper-bounds-4×4-asymptotics-etc.75180/#post-1333877).
On a 5x5, the move sequences R1 C0 R0 C0' and C0 R0 C0'...

5x5 NRG GN is at most 278.
I used the solving phases null --> LMN QRS --> LMNO QRST VWXY --> C HIJ LMNO QRST VWXY --> entire board. 25+65+82+106=278.
(Sidenote: for 4x4NRG, using the solving phases null --> IJKL MNOP --> entire board yields a bound of 28+94=122, which is worse than xyzzy's...

5x5 GN is at most 46.
I reduced the GN of 5x5: 0x0 --> 3x3 to 18, using several new techniques:
First, I split this solving stage into the phases null --> AC KM --> 3x3 (ABC FGH KLM) instead of null --> AB FG --> 3x3, which decreases the scrambles I would have to process from ~233e9 to ~188e9...

3-cycles and 2,2-cycles + Dijkstra's algorithm
This is similar to doing a standard Loopover BFS, except the transitions are 3-cycles and 2,2-cycles instead of single moves, and we have to use Dijkstra's algorithm instead of BFS (since we are now evaluating shortest paths in a weighted graph)...

A much better upper bound technique:
This one comes partially from xyzzy via a Discord post. As in regular Loopover, we can solve a NRG board by extending a block of solved pieces. Unlike in regular Loopover, when extending an existing block to a bigger block, the initial block must leave at...

New upper bound technique:
Let C3(a,b,c) be the minimum number of moves needed to cycle locations a-->b-->c-->a and C4(a,b,c,d) be the minimum number of moves needed to swap locations a<-->b and c<-->d.
Consider for simplicity only using the 3-cycle. Suppose the piece at the location (0,0) is...

New upper bounds:
I added permutation "symmetries" to the 3-cycle and double-swap algorithms. Code is at https://github.com/coolcomputery/Loopover-NRG-Upper-Bounds.
Basically, every algorithm A describes an action (L,P) for some array L of locations on the NxN board and a permutation P, where...

5x5 GN is at most 49.
I reduced the GN for 0x0->2x2->3x3 (i.e. solving AB FG, then C H KLM) to at most 21 moves., by considering every scramble of the pieces ABC FGH KLM that would take >21 moves by naively applying solving the two phases, applying at most 2 prefix moves, and solving the two...

6x6 GN is at most 101.
I reduced the GN of 4x5->5x5 + 5x5->6x6 to 38, by applying at most 5 prefix moves to every scramble that would naively take >38 moves. There were 249347110 total scrambles, which took 4294.610 seconds to process (including making the two BFS trees). Referring to this...

Oops, I had F gripped instead of A (it's the default gripped piece on https://loopover.xyz/). Here's the scramble again, this time with A gripped:
ABCD FBCJ
EFGH --> EPGL
IJKL NMIA
MNOP KHDO

New lower bounds:
Using a similar technique by xyzzy for regular Loopover, we can describe any NRG scramble as a list of "syllables", where a syllable is a run of Ds, Rs, Ls, or Us, and syllables alternate between horizontal (Rs and Ls) and vertical (Ds and Us). Then, for a NxN board, let S be...

Loopover NRG is like Loopover, but you can only "grip" a certain fixed piece at all times, i.e. you can only slide the row or the column that that piece is in. Here is an example solve (not by me).
Although there's been research on God's number for regular Loopover, I haven't found any for...

4x4 STM GD is at most 22
I added row reflection, column reflection, and transposition (flipping along the main diagonal) to my previous algorithm on improving the 0x0->2x3 and 2x3->4x4 block-building.
A total of 1647318090782 scrambles that required >=23 moves when the block-building was applied...

5x5 STM GD is indeed at most 53
I improved the 0x0->2x3 and the 2x3->3x4 steps differently: instead of solving translated regions of each scramble (which requires testing over all permutations of where the non-3x4 pieces can go in a 5x5), I added 1 prefix move before running the block-building...

4x4 STM GD is at most 23
https://github.com/coolcomputery/Loopover-Brute-Force-and-Improvement/blob/master/LoopoverBruteForce.java
The solved 4x4 board will look like this:
0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
1. Enumerate all ways of building the 2x3 whose upper-left corner is 0, 0...