Springer-resolution Hochschild-center conjecture for affine Springer-fiber cohomology

Let $P\subset G$ be a parabolic subgroup, let $\widetilde{N}=T^*(G/P)$ be the corresponding Springer resolution, and let $T\widetilde{N}$ be its tangent bundle. The dilation action of $\mathbb C^*$ gives the superscript grading $k$. Let $\tau$ be the Poisson bivector field on $\widetilde{N}$, and let $\Fl^{\gamma}_{\bI}$ be the affine Springer fiber with lattice action by $\Lambda$. Springer-resolution center conjecture. There is a bigraded algebra isomorphism

$$\bigoplus_{i+j+k=0}H^i(\widetilde{N},\wedge^jT\widetilde{N})^k\cong H^*(\Fl^{\gamma}_{\bI})^{\Lambda}.$$

Alternatively, there is a bigraded algebra isomorphism

$$\bigoplus_{i+j+k=0}H^i(\widetilde{N},\wedge^jT\widetilde{N})^k\cong \operatorname{gr}^P H^*(\Fl^{\gamma}_{\bI})^{\Lambda}.$$

Moreover, under this correspondence $\tau$ should correspond up to a scalar to the polynomial $\Delta_{(n-1,1)}$, or equivalently in the second version to $c_1(\mathcal L_{\det})$.

The conjecture links the Hochschild-theoretic description of singular quantum-group centers with affine Springer-fiber cohomology and its perverse grading. The source gives no resolution.