Stable Allen–Cahn level-set curvature conjecture in dimensions at most seven

Let $\Omega \subset \mathbb{R}^n$ be a domain, and let $u_{\varepsilon_j}$ be stable solutions of the Allen–Cahn equation in $\Omega$, where $\varepsilon_j \downarrow 0$. Assume that the sequence has uniformly bounded energy. The interfacial regions are $\{ -0.9 < u_{\varepsilon_j} < 0.9 \}$.

Stable Allen–Cahn regularity conjecture. If $n \leq 7$, then the curvatures of the level sets of $u_{\varepsilon_j}$ in these interfacial regions are uniformly bounded along the sequence on compact subdomains of $\Omega$.

A positive answer would give smooth, possibly multiplicity-bearing convergence of stable Allen–Cahn solutions to minimal hypersurfaces in the singular limit. The conjecture is presented as a fundamental regularity question and its resolution is linked to multiplicity-one and Morse-index questions for Allen–Cahn approximations.