Oblomkov–Shende conjecture for the HOMFLY polynomial of a curve singularity

Let $C$ be a locally planar curve, let $p\in C$, and let $c$ be the analytic germ of $C$ at $p$. Suppose that $c$ has $b$ branches and contributes $\delta$ to the arithmetic genus. Let $P_\infty$ denote the coefficient of the lowest power of $a$ in the HOMFLY polynomial, and let $n_h(c)$ be the Hilbert-scheme invariants of the germ. Then

Oblomkov–Shende conjecture.

$$P_\infty(\operatorname{Link}(C,p)) = \sum_{h=0}^\delta n_h(c) z^{2h-b}.$$

This conjecture relates the topology of the link of a locally planar curve singularity to Hilbert schemes of points on the curve germ. The source supplies no evidence of a resolution, so its status is left open.