The local characterization of f-convex drawings

Let $D$ be an h-convex drawing of $K_n$, meaning that every induced subdrawing on $K_5$ is convex, and let $D[J]$ denote the subdrawing induced by an isomorph $J$ of $K_8$. The local f-convexity conjecture. $D$ is f-convex if and only if, for every isomorph $J$ of $K_8$, $D[J]$ is f-convex. This proposes a forbidden-drawing characterization of f-convexity within h-convex drawings; the source gives examples of h-convex drawings that are not f-convex but provides no resolution of the conjecture.