The Oblomkov–Rasmussen–Shende conjecture for Hilbert schemes of points on singular curves

Let $C\subset\mathbb{C}^2$ be a singular curve with a planar singularity at the origin, and let $L$ be the algebraic link obtained by intersecting $C$ with a sufficiently small $3$-sphere centered at the origin. Write $\mathrm{Hilb}^n(C,0)$ for the Hilbert scheme of length-$n$ subschemes of $C$ supported at the origin, and let $\mathrm{HHH}(L)$ denote the colored Khovanov–Rozansky homology of $L$. Oblomkov–Rasmussen–Shende conjecture. The homology of the Hilbert schemes is identified with the homology of the corresponding link, namely

$$\mathrm{HHH}^{a=0}(L)=\bigoplus_{k,n=0}^{\infty}\mathrm{H}^k(\mathrm{Hilb}^n(C,0)).$$

This conjecture connects invariants of planar curve singularities with colored Khovanov–Rozansky homology of algebraic links. The supplied text states the conjecture but gives no evidence that it has been resolved.