Shende's conjecture relating braid varieties and affine Springer fibers

Let $C$ be an irreducible plane curve singularity, let $L$ be the corresponding algebraic knot, and let $\beta$ be the corresponding braid on $n$ strands. Let $X(\beta\Delta;w_0)$ be the associated braid variety, let $(\mathbb{C}^*)^{n-1}$ act as in the source, and let $\operatorname{Sp}_{Y}$ denote the corresponding affine Springer fiber. Shende's conjecture. There is an isomorphism

$$H_c^*\left(X(\beta\Delta;w_0)/(\mathbb{C}^*)^{n-1}\right)\simeq H^*(\operatorname{Sp}_{Y}),$$

under which the halved weight filtration on the left-hand side matches the perverse filtration on the right-hand side. This predicts a geometric correspondence between braid varieties, affine Springer fibers, and Hilbert schemes; the supplied text does not state whether the conjecture has been resolved.