The y-ified KhR-vs-Quot conjecture

Let $L_f$ be the link of the plane curve germ $f=0$. Let $\widetilde{\operatorname{Quot}}^{\vec{x},\vec{y}}$ be the bigraded $\mathbb{C}[\vec{x},\vec{y}]$-module assembled from the modified equivariant Borel--Moore homologies of the parabolic Quot schemes, and let $\Psi$ be the categorified functor described in the setup. $y$-ified KhR-vs-Quot conjecture. The $y$-ified Khovanov--Rozansky homology of $L_f$ is isomorphic, as a triply graded $\mathbb{C}[\vec{x},\vec{y}]$-module, to

$$\Psi\bigl(\widetilde{\operatorname{Quot}}^{\vec{x},\vec{y}}\bigr),$$

after appropriate regrading. This is a refinement of KhR-vs-Quot with no direct Hilbert-scheme analogue in the cited ORS formulation. The source does not establish it in general.