The $h$-convex drawings conjecture on crossed edges

Let $n\geq 11$. An $h$-convex drawing is a simple drawing satisfying the paper's $h$-convexity condition.

$h$-convex crossed-edges conjecture. There exists an $h$-convex drawing of $K_n$ in which every edge is crossed.

The conjecture extends the computationally observed examples for $n=11,\ldots,21$. It concerns the existence of convexity-restricted drawings with no uncrossed edge and remains open for all $n\geq 11$.