The ADE quantum field theory algebra and symplectic reflection algebra conjecture

Let $X$ be a resolution of an ADE singularity, with deformation $\mathscr O_{\epsilon,\lambda_1,\dots,\lambda_r}(X)$ of its algebra of functions, where $r$ is the ADE rank, and let $U^{QFT}_\delta(\mathscr O_{\epsilon,\lambda_1,\dots,\lambda_r}(X)\otimes\mathfrak{gl}_N)$ be the deformation obtained from the five-dimensional quantum field theory. Let $K$ be the charge of framed torsion-free sheaves. ADE large-charge conjecture. The algebra $U^{QFT}_\delta(\mathscr O_{\epsilon,\lambda_1,\dots,\lambda_r}(X)\otimes\mathfrak{gl}_N)$ is the large-$K$ limit of a deformation quantization of the moduli of framed torsion-free sheaves of charge $K$ and rank $N$ on the noncommutative deformation of $X$. For $N=1$, it should be the large-$K$ limit of the symplectic reflection algebra for the wreath-product action $S_K\ltimes\Gamma^K$ on $\mathbb C^{2K}$, where $\Gamma\subset SU(2)$ is the finite subgroup associated to the ADE singularity. This conjecture proposes an algebraic realization of the five-dimensional theory through moduli-space quantizations; proving it requires constructing and identifying the relevant large-charge limits.