Equivalence conjecture for the image of $\Phi_{y,A}$

Let $A$ be a subset of the affine simple reflections and let $y$ be an element of ${}^A W_{\mathrm{ext}}^S$; denote by $\mathsf{C}_{y,A}$ the Serre subcategory of $\mathsf{Perv}_{(I_{\mathrm{u}}^A,\mathcal{X}_A)}(\mathrm{Gr},\Bbbk)$ generated by the simple objects $\mathsf{L}^A_{yt_\lambda}$ with $\lambda \in -\mathbf{Y}_+$. Let $\Phi_{y,A}$ be the fully faithful exact functor from $\mathsf{Perv}_{\mathcal{L}^+G}(\mathrm{Gr},\Bbbk)$ to $\mathsf{C}_{y,A}$ described in the source. Image equivalence conjecture. If $y$ is minimal in ${}^A W_{\mathrm{ext}}^S$ for the Bruhat order, then

$$\Phi_{y,A}: \mathsf{Perv}_{\mathcal{L}^+G}(\mathrm{Gr},\Bbbk) \longrightarrow \mathsf{C}_{y,A}$$

is an equivalence of categories. This conjecture concerns when the fully faithful functor constructed in the geometric Steinberg setting is essentially surjective; the supplied text gives no resolution status.