$C^{1,1}$ regularity conjecture for the minimal Bellman function

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{B}_\varepsilon$ denote the minimal function associated with $f$ on the domain $\Omega_\varepsilon$. $C^{1,1}$ regularity conjecture. The function $\mathbb{B}_\varepsilon$ is of class $C^{1,1}$. The conjecture is motivated by known $C^{1,1}$ regularity results for the homogeneous Monge–Ampère equation on strictly convex domains with smooth boundary data, while the free-boundary setting here remains unresolved. The source notes that the smoothness and boundedness assumptions on $f$ may be stronger than necessary.