Tilting-bundle equivalence for the heart of the exotic t-structure on Grassmannians

Let $\mathbb{G}(k,N)$ be the Grassmannian, let $\mathcal{D}^b(\mathbb{G}(k,N))$ carry the exotic t-structure $(\mathcal{D}^{\geq 0}_{ex}(k,N-k),\mathcal{D}^{\leq 0}_{ex}(k,N-k))$, and let $\mathcal{T}:=\bigoplus_{\lambda}\mathbb{S}_{\lambda}\mathcal{V}$ be the tilting bundle given by the Kapranov exceptional collection. Set $A:=\operatorname{End}(\mathcal{T})$. The derived equivalence is

$$R\operatorname{Hom}(\mathcal{T},-):\mathcal{D}^b(\mathbb{G}(k,N))\xrightarrow{\cong}\mathcal{D}^b(A^{op}\text{-mod}).$$

Tilting-bundle equivalence conjecture. Under this derived equivalence, the restriction of the functor to the heart of the exotic t-structure gives an equivalence of abelian categories:

$$R\operatorname{Hom}(\mathcal{T},-):\mathcal{D}^{\geq 0}_{ex}(k,N-k)\cap\mathcal{D}^{\leq 0}_{ex}(k,N-k)\xrightarrow{\cong}A^{op}\text{-mod}.$$

The conjecture would identify the heart of the exotic t-structure with the abelian category of right modules over the endomorphism algebra of the Kapranov tilting bundle. The supplied text does not state whether this expected equivalence is known or remains open.