Subdivision conjecture for strictly convex vertex and edge-midpoint drawings

Let $G$ be a graph, and let subdivision mean a graph obtained by replacing edges by paths, possibly subdividing edges more than once. Let $\mathcal{G}_s^s$ denote the class of graphs admitting a straight-line drawing whose vertices and edge midpoints are both strictly convex.

Subdivision conjecture. Every graph has a subdivision obtained by multiplying subdividing its edges such that the resulting graph belongs to $\mathcal{G}_s^s$.

The conjecture is motivated by examples in which subdivisions of $K_{2,n}$ and $K_{3,3}$ admit such drawings even though the original graphs do not. The supplied text gives no resolution.