Hall's conjecture on Yang–Mills convergence for powers of simple loops

Let $(M,a)$ be a closed, area-weighted topological map, embedded in a closed, connected, orientable surface, and let $G_N=\mathrm{U}(N)$. A loop is included in a disc if its drawing is contained in an open, contractible set of the surface. For a simple loop $s$ included in a disc and an integer $n\geq 0$, write $\ell=s^n$, and let $W_\ell$ be the corresponding Wilson loop under $\mathrm{YM}_{M,a,G_N}$. Hall's conjecture. Whenever $\ell=s^n$ with $n\geq 0$ and $s$ is a simple loop included in a disc, $W_\ell$ converges in probability towards a constant as $N\to\infty$ under $\mathrm{YM}_{M,a,G_N}$. The conjecture removes the hypothesis used in the preceding theorem that convergence be assumed for these loops: it asserts that the required convergence holds for every power of every simple loop included in a disc. The source does not report a resolution, so the conjecture is treated as open.