Oblomkov–Shende conjecture for HOMFLY polynomials of algebraic links

Let $C$ be a curve in $\mathbb{C}^2$ given by $f(x,y)=0$ and passing through the origin. Intersecting $C$ with the boundary of a sufficiently small ball around the origin gives an oriented link in the $3$-sphere. Let $C^{[n]}_0$ be the moduli space of pairs $(F,s)$ with $F$ a torsion-free sheaf on $C$, $s:{\mathcal O}_C\to F$ a section, and $\dim F/s{\mathcal O}_C=n$, whose section vanishes only at the origin. Let $C^{[n];m}_0$ be the locus where $m=\dim_{\mathbb C}F/(x,y)F$, and let $\mu=\dim_{\mathbb C}\mathbb{C}[[x,y]]/(\partial_xf,\partial_yf)$ be the Milnor number. If $H(L)$ denotes the HOMFLY polynomial of the link $L$, then Oblomkov–Shende conjecture.

$$H(\text{the link of }C)=\left(\frac{Q}{q}\right)^{\mu-1}\sum_{n,m}q^n(1-Q)^m\chi(C^{[n];m}_0).$$

The source explains that this conjecture was proven mathematically for torus knots, while presenting the surrounding argument as a physics proof. Thus the general statement is treated as open here, with known special cases.