Localization conjecture comparing categorifications of the quantum open unipotent cell

Fix a positive integer $N$, set $w=(s_0s_1)^N$, and let $\widetilde{\mathcal{D}}^\sharp_w$ be the localization of the graded monoidal category $\mathcal{D}^\sharp_w$ by the objects $\mathrm{C}_0$ and $\mathrm{C}_1$. Let $\mathcal{P}\mathcal{C}oh([\operatorname{Gr}/G_N^c(\mathcal{O})])$ be the monoidal category of equivariant perverse coherent sheaves on the affine Grassmannian, with convolution product $\circledast$, and let $\widetilde\Psi_w$ be the induced graded functor.

Localization equivalence conjecture. The functor $\widetilde\Psi_w$ is a graded monoidal equivalence

$$\big(\widetilde{\mathcal{D}}^\sharp_w,\,\circ\big)\to\big(\mathcal{P}\mathcal{C}oh([\operatorname{Gr}/G_N^c(\mathcal{O})]),\,\circledast\big).$$

This conjecture compares two monoidal categorifications of the quantum open unipotent cell associated with $w$. The source constructs a faithful functor and an isomorphism on Grothendieck groups, but full equivalence is not established.