Output-polynomial enumeration for $H$-free incomparability graphs

Let $H$ be a graph. It is co-bipartite if its vertex set can be partitioned into two cliques, and let $\textsc{Dom-Enum}$ denote the problem of enumerating all minimal dominating sets of a graph.

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.

This is described as a bold generalization of the preceding conjecture. The case $H=C_4$ is known for incomparability graphs because $C_4$-free incomparability graphs are interval graphs, but the conjecture for arbitrary co-bipartite $H$ remains open.