I have been trying to analyse the game of Ordinal Chomp played on a $3 \times 3 \times \omega$ board. The rules can be found in the Wikipedia article, briefly:

- This game is played between two players on the set $3 \times 3 \times \omega$. A move is to pick any remaining $(i,j,k)$ and remove all $(i^\prime,j^\prime,k^\prime)$ where $i^\prime \geq i$, $j^\prime \geq j$ and $k^\prime \geq k$. The player to take $(0,0,0)$ loses.

Unfortunately, the analysis is extremely complicated, so I have been unable to find a winning move for the first player or a proof that none exists. So, my question is:

Is there a winning move in $3 \times 3 \times \omega$ Ordinal Chomp for the first player, and if so, what is it?