I'm almost certain that this is an existing method, but I can't remember which one it isPossible new Roux-based 4x4 method?
1. First 2 centers
2. First Block on one of the solved centers, but with additional layer (3x4x2 block)
3. Second Block (just outer layer, so 3x4x1)
5. Solve the rest of the cube using U and the right inner slice (still working on this part)
Essentially, the point of this method is to eliminate the numerous regrips that Lewis/Stadler has when switching between wide slice, left inner slice, and right inner slice moves. Roux's obvious weakness on big cubes is not being able to keep the inner layers together during 3x3 stage, and this is an attempt to solve that as well. There's also probably a way to avoid OLL parity by orienting all the edge-halves during step 5, but I haven't fully fleshed that out yet.