Perverse-filtration and Cherednik-filtration conjecture for compactified Jacobians

For coprime positive integers $k,n$, let $L_{k/n}$ be the finite-dimensional simple representation of the rational Cherednik algebra with parameter $k/n$, let $eL_{k/n}$ be its spherical part, and let $\Lambda^{i,j}$ denote the filtration of Proposition CEE. Let the cohomology of the compactified Jacobian of the singularity $X^n=Y^k$ carry its perverse filtration. Perverse-filtration conjecture. The perverse filtration and the filtration $\Lambda$ of Proposition CEE agree.

This identification would connect the geometric perverse filtration with the representation-theoretic filtration used to compute torus-knot homology. The source gives no resolution status.