The $P=W$ conjecture for positroid torus quotients and compactified Jacobians

Let $\Pit_{k,n}$ be the torus quotient and let $J_{k,n-k}$ be the compactified Jacobian of the plane curve singularity $x^k=y^{n-k}$. Their cohomology carries a weight filtration and a perverse filtration, respectively. The $P=W$ conjecture. There is a deformation retraction from $\Pit_{k,n}$ to $J_{k,n-k}$ sending the weight filtration of $H^\bullet(\Pit_{k,n})$ to the perverse filtration of $H^\bullet(J_{k,n-k})$. This conjecture is motivated by the relationship between compactified Jacobians, positroid varieties, and knot invariants, and arises from the $P=W$ phenomena studied for character varieties. The supplied text does not indicate whether this particular statement has been resolved.