Singer's master-field conjecture for Yang–Mills holonomies

Let $\Sigma$ be a closed, connected, orientable, two dimensional, Riemannian manifold, the Euclidean plane $\mathbb{R}^2$, or a disc in $\mathbb{R}^2$. Let $G_N$ be a classical group of size $N$, with metric given by the specified inner product on its Lie algebra. For a loop $\ell\in \mathrm{L}(\Sigma)$, let $W_\ell$ denote its Yang–Mills Wilson loop under $\mathrm{YM}_{\Sigma,G}$. Singer's master-field conjecture. For every loop $\ell\in \mathrm{L}(\Sigma)$, there is a constant $\Phi_\Sigma(\ell)$ such that

$$W_\ell\to\Phi_\Sigma(\ell)\quad\text{in probability as }N\to\infty.$$

Moreover, if $\Psi$ is a diffeomorphism of $\Sigma$ preserving its volume form, then

$$\Phi_\Sigma(\Psi(\ell))=\Phi_\Sigma(\ell).$$

This is a large-$N$ master-field conjecture for Yang–Mills theory: Wilson loops are expected to converge to deterministic limits depending only on the area-preserving geometry of the surface. The source states that the conjecture remains open for general surfaces.