The Yoneda–convolution conjecture for half-density sheaves

Let $\mathcal{S}_{n-k,k}$ be the Slodowy slice, let $Y$ be the corresponding Springer fiber, and let $A$ range over all irreducible components of $Y$. Write $i_A:A\hookrightarrow \mathcal{S}_{n-k,k}$ for the inclusion, and let $\mathcal{\Omega}(A)^{1/2}$ be the associated half-density sheaf. The Ext-algebra is equipped with the Yoneda product, while $H(\widetilde{Y}\times_{\mathfrak{g}}\widetilde{Y})$ is equipped with convolution. Yoneda–convolution conjecture. There is an isomorphism of algebras

$$\operatorname{Ext}^\bullet_{\operatorname{Coh}(\mathcal{S}_{n-k,k})}\Big(\bigoplus_A i_{A*}\mathcal{\Omega}(A)^{1/2},\bigoplus_A i_{A*}\mathcal{\Omega}(A)^{1/2}\Big)\cong H(\widetilde{Y}\times_{\mathfrak{g}}\widetilde{Y}^{}).$$

This would give an explicit description of the Ext-algebra of the half-densities by identifying its Yoneda multiplication with convolution; the source introduces it as a conjecture and gives no resolution.