The face-subgraph characterization conjecture for compact visibility graphs

Let $G$ be a planar graph. For each internal face $F$ in a plane drawing of $G$, let $G_F$ denote the subgraph induced by the vertices incident with $F$. Compact visibility conjecture. The graph $G$ is a compact visibility graph if and only if it has a plane drawing such that, for every internal face $F$, the graph $G_F$ is a compact visibility graph.

This would characterize compact visibility graphs through the compact visibility of the subgraphs induced by the boundaries of internal faces. The statement is motivated by the failure of the analogous characterization using only triangular internal faces, since $K_{1,1,n}$ is a compact visibility graph but has no plane drawing with all internal faces triangular for $n\geq 3$.