2-degeneracy conjecture for strictly convex vertex and edge-midpoint drawings

A point set is strictly convex if every point is a vertex of its convex hull. For a graph $G$, let $\mathcal{G}_s^s$ be the class of graphs admitting a straight-line drawing in the plane in which the vertices form a strictly convex point set and the midpoints of the edges also form a strictly convex point set.

2-degeneracy conjecture. If $G\in\mathcal{G}_s^s$, then $G$ is $2$-degenerate; that is, every non-empty induced subgraph of $G$ has a vertex of degree at most $2$.

The authors do not know any 3-regular graph in $\mathcal{G}_s^s$. The conjecture would imply strong structural and sparsity restrictions on graphs admitting these drawings, but the supplied text gives no resolution.