The Procesi-bundle conjecture for affine Springer fiber homology

Let $\mathcal{P}$ be the Procesi bundle on $\operatorname{Hilb}^n(\mathbb{C}^2)$, let $\mathcal{P}^*$ denote its dual, and let $\mathcal{O}(d)$ be the indicated determinant-line-bundle twist restricted to $\operatorname{Hilb}^n(\mathbb{C}^*\times\mathbb{C})$. Let $\Delta$ denote the grading shift appearing in the preceding identification, and let $\widetilde{\operatorname{Sp}}_{zt^d}$ be the affine Springer fiber in the affine flag variety. Procesi-bundle conjecture. As graded $\mathbb{C}[y_1,\ldots,y_n]$-modules, one has

$$H^0\bigl(\mathcal{P}\otimes\mathcal{P}^*\otimes\mathcal{O}(d),\operatorname{Hilb}^n(\mathbb{C}^*\times\mathbb{C})\bigr)\cong \Delta^d\cdot H_*^T\bigl(\widetilde{\operatorname{Sp}}_{zt^d}\bigr).$$

The conjecture proposes a sheaf-theoretic description of the equivariant homology of affine Springer fibers in the affine flag variety. The source explicitly says that it is not clear what this cohomology describes and provides no evidence of a resolution.