The Hilb-vs-Quot conjecture for plane curve germs

Let $R$ be the complete local ring of a complex algebraic plane curve germ and let $S$ be its normalization. For each $\ell\geq 0$, let $\mathcal{H}^{\ell}=\operatorname{Quot}^{\ell}(R)$ and $\mathcal{Q}^{\ell}=\operatorname{Quot}^{\ell}(S)$, and let $\chi(X,t)$ denote the virtual weight polynomial of a finite-type complex scheme $X$. Define

$$\operatorname{Hilb}(q,t)=\sum_{\ell\geq 0}q^{\ell}\chi(\mathcal{H}^{\ell},t),\qquad \operatorname{Quot}(q,t)=\sum_{\ell\geq 0}q^{\ell}\chi(\mathcal{Q}^{\ell},t).$$

Hilb-vs-Quot conjecture. For any plane curve germ,

$$\operatorname{Hilb}(q,t)=\operatorname{Quot}(q,q^{\frac12}t).$$

This conjecture relates the virtual weight polynomials of Hilbert schemes and Quot schemes and extends a conjecture of Cherednik to plane curve germs with multiple branches. It is proved in the paper for the germs $y^3=x^d$ with $3\nmid d$, but remains open in general.