Interior regularity conjecture for non-degenerate nonlocal mean field games

Let $2\sigma\in(0,2)$ and let $\mathcal{L}$ be a non-degenerate nonlocal operator satisfying condition (LC), let $F$ satisfy the non-degeneracy and structural assumptions, and let $(f,g)\in\mathcal{R}_B(\alpha,M)$. Consider the Hamilton–Jacobi–Bellman equation associated with the mean field game system. Interior regularity conjecture. Assume (LC), the non-degeneracy condition on $F$, $(f,g)\in\mathcal{R}_B(\alpha,M)$, and assumptions (a:F1) and (a:F2). Then interior $(\alpha/2\sigma,\alpha)$-regularity estimates hold for the Hamilton–Jacobi–Bellman equation. The conjecture concerns interior regularity for a broad class of nonlocal operators, including fractional Laplacians and nonsymmetric operators, in the fully nonlinear setting. The source notes that no results beyond the affine case are available.