The parabolic Hilb-vs-Quot conjecture

Assume $R=\mathbb{C}[[x]][y]/(f)$ defines a generically separable degree-$n$ cover of the $x$-axis, fully ramified at the origin. For an integer composition $\nu$ of $n$, let $\mathcal{H}_\nu^\ell$ and $\mathcal{Q}_\nu^\ell$ parametrize a module $M$ of colength $\ell$ together with a $y$-stable flag on $M/xM$ of parabolic type $\nu$, and let $\operatorname{Hilb}_\nu(q,t)$ and $\operatorname{Quot}_\nu(q,t)$ be the corresponding generating series. Parabolic Hilb-vs-Quot conjecture. For any $R$ and $\nu$ as above,

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

This is a parabolic refinement of Hilb-vs-Quot and is designed to compare the Hilbert- and Quot-scheme formulations of the Oblomkov--Rasmussen--Shende conjecture. The paper proves it for $f(x,y)=y^3-x^d$ with $d$ coprime to $3$, while the general statement remains open.