A revision to Ultimate TicTacToe

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There was a question about tictactoe*tictactoe (found here - A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe). But I was thinking that a variant could be made to make the game easier to solve. So, in the old version, when a square of the large tictactoe board is taken, you are not allowed to play in that square again. However, that makes the game much harder to solve. So I am proposing a variant where you are allowed to play on a square that has already been taken. If your move would also create a 3 in a row in that square, you will negate the first 3 in a row, and the square will become neutral, until a third 3 in a row can be made (which may actually happen). Thus, play will go on until all squares are filled, which will hopefully make solving the problem easier. So could someone make a game for this somewhere? And does this addition make the game easier to solve?
Thanks.