The enhanced y-ified KhR-vs-Quot conjecture

Let $\bar{Y}_f$ be the enhanced $y$-ified Khovanov--Rozansky module of the link $L_f$. Let $\widetilde{Q}_{S}^{\vec{x},\vec{y}}$ be the bigraded $\mathbb{C}[\vec{x},\vec{y}]$-module formed from the equivariant Borel--Moore homology of the full parabolic Quot construction, with its Springer action, and let $\Psi$ be the functor defined in the setup. Enhanced $y$-ified KhR-vs-Quot conjecture. In the setup above:

1. $\bar{Y}_f$ is supported in integral tridegrees.
2. There is an isomorphism of $\mathbb{C}[\vec{x},\vec{y}]$-modules

$$\bar{Y}_f\xrightarrow{\sim}\Psi(\widetilde{Q}_{S}^{\vec{x},\vec{y}})$$

that sends degree $(i,j,k)$ to degree $(i,j,2k)$. In particular, $\Psi(\widetilde{Q}_{S}^{\vec{x},\vec{y}})$ is supported in even cohomological degrees. This is the most precise action- and grading-sensitive form of the paper's KhR-vs-Quot proposal. The source gives no proof of the full statement, so it remains open.