Galashin–Lam's deformation-retract and filtration conjecture for open positroid varieties

Let $k$ and $n$ be coprime positive integers, let $X^\circ_{k,n}$ denote the top-dimensional open positroid variety, and let $J_{k,n-k}$ be the compactified Jacobian of the plane curve singularity $x^k=y^{n-k}$. A deformation retract is understood in the topological sense, without requiring it to be algebraic.

Galashin–Lam's conjecture. There exists a non-algebraic deformation retract

$$r:X^\circ_{k,n}\to J_{k,n-k}$$

from $X^\circ_{k,n}$ onto $J_{k,n-k}$. Moreover, in compactly supported cohomology,

$$r^*:H_c^*(J_{k,n-k})\to H_c^*(X^\circ_{k,n})$$

sends the weight filtration to the perverse filtration.

This conjecture connects the topology and filtrations of the open positroid variety with those of the compactified Jacobian of a plane curve singularity. The supplied text attributes the motivation to work of Galashin–Lam and to the works cited as STZ17 and STWZ19, but gives no resolution status.