The KhR-vs-Quot conjecture

Let $f(x,y)=0$ be a plane curve germ as above, let $\bar{X}_f(a,q,t)$ be the normalized Khovanov--Rozansky homology polynomial, and let $\Psi$ be the map from the relevant symmetric-function representation to the $a$-graded polynomial. Let $\mathcal{F}\operatorname{Quot}(q,t)$ denote the symmetric-function-valued generating series for the parabolic Quot schemes. KhR-vs-Quot conjecture. For any such $f$,

$$\bar{X}_f(a,q,t^2)=\Psi\bigl(a,\mathcal{F}\operatorname{Quot}(q,t)\bigr).$$

This is the Quot-side reformulation of the ORS conjecture, conditional on the parabolic Hilb-vs-Quot conjecture. Its general validity is open, although the paper proves it in the stated $y^3=x^d$ family with $3\nmid d$.