No idea. Hmm, off the cuff maybe I’d conjecture that there cannot exist any initial configurations such that the ant gets stuck in a loop of finite length for infinite iterations.

On any Nth iteration, the ant has only touched <= N tiles. Suppose we do a thought experiment where we create a second board that is empty except for the initial states of the touched tiles up to the Nth step. This second board has finite support, and so the usual theorems kick in, and consequently the ant will eventually start building a highway that would carry it away from the initial loop.

I’m just making up guesses though, I have no clue if that is right.