Convexity conjecture for the spectral separation utility

Let $\mathcal{U}:[0,\infty)\to\mathbb{R}$ be the utility function measuring the separation between the connected components of the support of the limiting spectral distribution $F_t$; in particular, when the support has two connected components $A_t$ and $B_t$, set

$$\mathcal{U}(t)=\min_{a\in A_t,\,b\in B_t}|a-b|.$$

Utility-function conjecture. $\mathcal{U}$ is non-increasing and convex in $t$. Numerical evidence supports this stronger claim, which would describe not only when the information/noise spectral gap disappears but also how the gap changes with the noise power. The paper does not provide a proof.