Convexity conjecture for the maximal limit shape

Let $F_{\rm \max}$ be the maximal value function whose derivative appears as the limit shape in the paper's limit-shape corollary, and let $F_{\rm \max}'$ denote that limit shape. Convexity conjecture. The limit shape $F_{\rm \max}'$ is always convex. This is posed as an open question based on computer-generated limit shapes for various density domains; no proof is supplied in the source.