The perverse filtration comparison conjecture for compactified Jacobians

Let $C_{q/p}$ be the plane curve under consideration, let $J_{q/p}$ be its compactified Jacobian, and let $P$ be the perverse filtration on $\textup{H}^{*}(J_{q/p})$. Let $M^{rig}_1(\mathbb{P}^1,C_{q/p})$ be the moduli space of rigidified maps, and let $\mathcal{O}_{q/p}$ be its local Artinian coordinate ring with residue field $\mathbb{Q}$, graded by the induced $\mathbb{G}_m$-action. Write $\mathfrak{m}$ for its maximal ideal. The perverse filtration comparison conjecture. For every $i,j\real$ there is a canonical isomorphism

$$\operatorname{Gr}^{P}_{j}\operatorname{H}^{2i}(J_{q/p})\cong \operatorname{Gr}^{j-i}_{\mathfrak{m}}\mathcal{O}_{q/p}[j].$$

These isomorphisms are compatible with the ring structures on both sides. This predicts a common degeneration of the cohomology ring and $\mathcal{O}_{q/p}$, but no resolution is supplied in the source.