$C^{1,1}$ regularity conjecture for the functions $\mathbb{U}$ and $\mathbb{V}$

Let $f$ be a smooth uniformly bounded function, and let $\mathbb{U}$ and $\mathbb{V}$ be the two functions associated with $f$ in the paper's Bellman-function formulation. $C^{1,1}$ regularity conjecture for $\mathbb{U}$ and $\mathbb{V}$. The functions $\mathbb{U}$ and $\mathbb{V}$ are of class $C^{1,1}$. This is the second smoothness conjecture proposed after comparison with regularity results for homogeneous Monge–Ampère boundary value problems. The source gives no evidence that either conjecture has been resolved.