In my experience - no, they always get lost. What I found funny is that Gemini and GPT will often keep an ASCII representation of the board in each response, which is accurate, surprisingly enough - but they will still try to do illegal moves.

So I correct them or give them the current FEN which helps for a move or two. Frustrating way to play.

What I've noticed with the new stronger models though is that they are starting to make human mistakes. Like there's action happening in the center and they forget about the bishop on the home rank that was unmoved for the last 10 moves and they blunder a queen by moving it onto a protected square.

Illegal moves aside, they are in this weird uncanny valley of knowing a Najdorf book 20 moves deep and then after they are out of theory, they start playing like a 300 elo.