Affine Springer–Hilbert-scheme sheaf correspondence

For $G=GL_n$, let $gamma\bin g derbrace{\llbracket t\rrbracket}_{\mathcal O}$ be regular semisimple, let $cbeta=\beta(gamma)$ be the associated braid, and let $i_{cC^*}:{cC}^*\times{cC}\to{cC}^2$ be the inclusion $(x,y)mapsto(x-1,y)$. Write $cF_gamma$ for the affine-Springer sheaf and $cG_cbeta$ for the corresponding complex on $cHilb^n({cC}^2)$. Affine Springer–Hilbert-scheme correspondence. There is an isomorphism of ${cC}^*$-equivariant sheaves

$$cF_\gamma\simeq i_{{cC}^*}^*(cG_cbeta),\qquad \beta=\beta(\gamma).$$

The conjecture is known at the level of Euler characteristics in the elliptic $SL_n$ case, but the sheaf-level isomorphism remains open.