I've recently come back to investigating ordinal chomp. See A winning move for the first player in $3 \times 3 \times \omega$ Ordinal Chomp for a definition. I made a new discovery, that the position \begin{pmatrix}\omega&\omega&1 \\ \omega&\omega&1 \\ 2&0&0\end{pmatrix} is a losing position, first by a computer search, then by a proof showing that all possible moves from this position are winning (I can outline the proof on request). It seems like positions like this (I call them quadruple $\omega$ positions) are extremely rare, and I haven't been able to find another example so far.
My question is: Are there other quadruple $\omega$ positions in $3 \times 3 \times \omega$ chomp, and if so, what are they?
