A Pairing Strategy That Almost Works

Continues from my previous short form. https://yoonwoojeong.github.io/SYZYGY Here is a small puzzle I developed. Two players each control one piece on a square grid. On your turn you either take a step, or you knock out one of the squares next to you so nobody can ever use it again. You can't walk over your own footprints, and if you can't move, you lose. It's a bit like a game of tag played while the floor crumbles. Let's make the game precise. Picture an grid of squares (the game uses , with columns a–g and rows 1–7). White's piece starts on c2 and Black's on e6, and White moves first. On your turn you do one of two things: Step to a square right next to you (up, down, left or right), or Delete a square right next to you, for both players, forever. Who wins with with perfect play? Here's the natural trick for the second player. Whatever White does, Black does the half-turn-rotated version of it. White steps up? Black steps down. White deletes the square to the left of their piece? Black deletes the square to the right of theirs. On a board with an even side (4×4, 6×6) there is no such square. So Black copies forever, never runs out of moves before White does, and the second playe