Equivariant tilting-generator conjecture for affine Grassmannian convolution resolutions

Let $G$ be a connected reductive algebraic group, let $\operatorname{Gr}=G(\mathcal K)/G(\mathcal O)$ be its affine Grassmannian, and let $\lambda_1^\vee,\ldots,\lambda_k^\vee$ be minuscule coweights with $\lambda^\vee=\lambda_1^\vee+\cdots+\lambda_k^\vee$. Let

$$p\colon \operatorname{Gr}^{\lambda_1^\vee}\star\cdots\star\operatorname{Gr}^{\lambda_k^\vee}=\widetilde{\operatorname{Gr}}^{\underline{\lambda}^\vee}\longrightarrow\overline{\operatorname{Gr}}^{\lambda^\vee}$$

be the convolution resolution. If $\operatorname{Gr}^{\omega}$ is the closed $G(\mathcal O)$-orbit in $\overline{\operatorname{Gr}}^{\lambda^\vee}$, and $\widetilde{\mathcal W}_{\omega}^{\lambda^\vee}$ and $\mathcal W_{\omega}^{\lambda^\vee}$ are respectively the preimage and transversal slice described in the source, write $p'$ for the induced conical symplectic resolution. Equivariant tilting-generator conjecture. There is a $G(\mathcal O)\rtimes\mathbb G_m$-equivariant sheaf $\mathcal E$ on $\widetilde{\operatorname{Gr}}^{\underline{\lambda}^\vee}$, whose restriction $\mathcal E'$ to $\widetilde{\mathcal W}_{\omega}^{\lambda^\vee}$ is a tilting generator for $p'$. The conjecture proposes a global equivariant extension of a tilting generator on the transversal-slice resolution; the supplied text presents it as a proposal and gives no resolution status.