The differential-operator description of spherical operators for holomorphic twists

Let $G$ be a reductive algebraic group. Let $R_2=H^\bu(\mathbb{A}^2,\mathcal{O}_{\mathbb{A}^2}^{\mathrm{alg}})$, let $\mathfrak{g}$ be the Lie algebra of $G$, and let $B(\mathfrak{g}\otimes R_2)$ denote the corresponding formal moduli space. Consider $\rho_*\mathcal{O}_{\mathrm{fp}}$, the algebra of spherical operators for the holomorphically twisted theory, equipped with the anti-diagonal filtration. Differential-operator conjecture. There is a map from the algebra of differential operators on $B(\mathfrak{g}\otimes R_2)$ to the second page of the spectral sequence for the anti-diagonal filtration on $\rho_*\mathcal{O}_{\mathrm{fp}}$. This gives a precise, weaker form of the expected description of the deformation of the dg Weyl algebra of spherical operators; the source does not provide evidence resolving whether the asserted map exists.