Domino Drift - Cool, Sophistic

Domino Drift is a fun game to interact with, based on the classic known car parking game rush hour, and a difficult mathematical problem.
Just pick a board size and start drifting the magic tiles!

You face a squared board, that is almost fully covered by domino stones. The only free place is the top-left corner.
You can slide a domino tile to the free block and reveal another block freely. Your target is to reveal the four corners by swiping the domino stones.

Consider yourself smart among us? Try to prove that every initial board is solvable.