The way I have always looked at this is that to know whether a square is attacked by the opponent during your turn you should pass that turn, and see whether the opponent then can move there. So the question then is what the turn-pass would do for the movement capabilities of the Joker. There are several cases that could be argued for:
The Joker keeps its move from the previous turn, because no piece was moved during the turn pass, leaving the 'last-moved piece' just as it was. In Greg's diagram black would be stalemate if the Joker had moved to b2 through a Knight move (e.g. imitating a black Knight that just moved to b2 while capturing it). But checkmate if it had been capturing a Bishop that just moved there, as it would then still be imitating that Bishop, and thus check black.
The turn pass is considered a move of the King (like in Chu Shogi it would be a move of the Lion). The Joker would then always check like a King during the opponent turn. In Greg's diagram black would always be be checkmated, even if it had moved there through a Knight move (e.g. imitating the Knight that was on b2 while capturing it).
The Joker must imitate the null move. It would then not be able to deliver check at all, and the position would always be stalemate. (During its own turn it could still capture a King, though, through whatever move it has then.)
The way I have always looked at this is that to know whether a square is attacked by the opponent during your turn you should pass that turn, and see whether the opponent then can move there. So the question then is what the turn-pass would do for the movement capabilities of the Joker. There are several cases that could be argued for: