Diagonal-coinvariant conjecture for the perverse-graded affine Springer-fiber cohomology

Let $G=GL_n$, let $\gamma$ be the regular element used to define the affine Springer fiber $\Fl^{\gamma}_{\bI}$, let $\Lambda$ be the lattice acting on it, and let $P$ be the perverse filtration. Let $\DR_n$ denote the associated diagonal coinvariant object. Diagonal-coinvariant conjecture. As bigraded vector spaces,

$$\DR_n\cong \operatorname{gr}^P H^*(\Fl^{\gamma}_{\bI})^{\Lambda}.$$

The claim is known in the source for $G=SL_2$. It is intended to identify the bigraded affine Springer-fiber cohomology with the diagonal coinvariant ring, but no general resolution is supplied.