Since the object of the game is checkmate or stalemate, the game would end before that could happen. If a Brain is on a space whose arrows do not point toward the opponent’s side, this would allow the other Brain to move to the same column without anything between them. And if that Brain moved to a space with an arrow pointing toward the Brain, it could check it. If the player with a checked Brain could not escape check, it would be checkmate, and the game would end.
Since the object of the game is checkmate or stalemate, the game would end before that could happen. If a Brain is on a space whose arrows do not point toward the opponent’s side, this would allow the other Brain to move to the same column without anything between them. And if that Brain moved to a space with an arrow pointing toward the Brain, it could check it. If the player with a checked Brain could not escape check, it would be checkmate, and the game would end.