Large-\N critical values for quantum minimal surfaces in flat tori

Let $M$ be a symplectic surface and let $S_{cl}$ be the classical functional for maps from $M$ to a flat torus $(S^1)^n$. For each $N$, let $\Phi_1,\dots,\Phi_n\in U(N)$ and let $S_q$ be the unitary matrix functional defined by the commutator terms. Large-$N$ critical-value conjecture. If $N\to+\infty$, limits of critical values of $S_q$ are either $+\infty$, or positive $\mathbb{Z}$-linear combinations of critical values of $S_{cl}$. Moreover, if the limit is finite, then $\|\Phi_i\Phi_j-\Phi_j\Phi_i\|\leq\operatorname{const}/N$. The conjecture predicts that finite-energy large-$N$ critical points become asymptotically commuting and that their critical values are assembled from classical minimal-surface values. The source gives no resolution.