After solving the top and bottom centers I've realized the regular 4x4 method won't work. It was impossible to track all the remaining center pieces and find their places. I was able to make some pairs, but due to the ghostly nature of the puzzle, I couldn't align and preserve the pairs while trying to make more pairs.
Change of plans. Solved the top and the bottom faces instead. Much easier and didn't take too long.
For the two middle layers I found a solved block of two pieces and started from there. Adding piece by piece, either it was a center piece, or an edge one. I was using commutators (three-cycles) and conjugates for that part.
What I am left with now is two double edges, and two and a half centers to solve yet. Going to complete the centers first and leave the edges for the very last part of the solve. I need to train three-cycling the edges on a regular 4x4 first.
The puzzle is hard to solve in one go. It's not the time, but the focus it requires. No big mistakes so far! I spent on it about 6 hours maybe. I am almost accustomed to it being too loose now.
Update:
It's been solved! And will stay so for a while.
The left side shows the solved state before reversing setup moves. This gave me a little scare.
While solving the last stages I encountered both edge parities. And a situation with only two center pieces being unsolved, which is impossible on a 4x4. It was a false equivalency. The bottom center is symmetrical, and it was rotated 90 degrees.
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