  To each complex reflection group $\Gamma$ one can attach a canonical
symplectic singularity $\mathcal{M}_\Gamma$ arXiv:math/9903070. Motivated by
the 4D/2D duality arXiv:1312.5344, arXiv:1707.07679, Bonetti, Meneghelli and
Rastelli arXiv:1810.03612 conjectured the existence of a supersymmetric vertex
operator superalgebra $\mathsf{W}_\Gamma$ whose associated variety is
isomorphic to $\mathcal{M}_\Gamma$. We prove this conjecture when the complex
reflection group $\Gamma$ is the symmetric group $S_N$ by constructing a sheaf
of $\hbar$-adic vertex operator superalgebras on the Hilbert scheme of $N$
points in the plane. For that case, we also show the free-field realisation of
$\mathsf{W}_\Gamma$ in terms of $\operatorname{rk}(\Gamma)$ many $\beta\gamma
bc$-systems proposed in arXiv:1810.03612, and identify the character of
$\mathsf{W}_\Gamma$ as a certain quasimodular form of mixed weight and multiple
$q$-zeta value.
  In physical terms, the vertex operator superalgebra $\mathsf{W}_{S_N}$
constructed in this article corresponds via the 4D/2D duality arXiv:1312.5344
to the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory with
gauge group $\operatorname{SL}_N$.
