The Oblomkov–Rasmussen–Shende conjectural correspondence in affine Springer homology

Let $\gamma$ be an unramified element with associated link $L_{\gamma}$, let $\mathrm{Sp}_{\gamma}$ be its affine Springer fiber, and let $\mathrm{gr}_P$ be the specified perverse filtration. Oblomkov–Rasmussen–Shende conjecture. One has

$$\mathrm{gr}_P H_*(\mathrm{Sp}_{\gamma})\otimes\mathbb{C}[x]\cong\mathrm{HHH}^{a=0}(L_{\gamma}).$$

The statement has been proved for torus knots and for $(n,nd)$-torus links, while it is open in the stated generality.