The convex-extension conjecture for COM tope graphs

Let $\mathcal{G}_{\rm COM}$ and $\mathcal{G}_{\rm OM}$ denote the classes of tope graphs of complexes of oriented matroids and oriented matroids, respectively. A graph $G$ is a convex subgraph of a graph $G'$ if $G$ is an induced subgraph containing every shortest path in $G'$ between vertices of $G$.

Convex-extension conjecture. Every $G\in\mathcal{G}_{\rm COM}$ is a convex subgraph of a graph $G'\in\mathcal{G}_{\rm OM}$.

This restates a conjecture from the cited literature in terms of tope graphs. It asks whether every COM tope graph can be embedded convexly into an oriented-matroid tope graph; the supplied text does not indicate that the conjecture has been resolved.