3 problems

The co-bipartite forbidden-subgraph conjecture. For any co-bipartite $H$, there is an output-polynomial time algorithm for $\textsc{Dom-Enum}$ in $H$-free incomparability graphs.

The $2K_p$-free incomparability conjecture. For every $p$, there is an output-polynomial time algorithm for $\textsc{Dom-Enum}$ in $2K_p$-free incomparability graphs.

The $S_t$-free incomparability conjecture. For every $t$, there is an output-polynomial time algorithm for $\textsc{Dom-Enum}$ in incomparability graphs of $S_t$-free posets.