Oblomkov–Yun perverse and maximal-ideal filtration conjecture for compactified Jacobians

Let $m,n$ be coprime positive integers, let $C_{\frac{m}{n}}$ be the closure of $y^m=x^n$ in $\mathbb{P}^2$, and let $J_{\frac{m}{n}}$ be its compactified Jacobian. Let $P$ be the perverse filtration on its cohomology, and let $\mathcal{M}_{\frac{m}{n}}$ be the arc space described in the source, with coordinate ring filtered by powers of its maximal ideal $\mathfrak{m}$. Write $\operatorname{gr}_j^P$ for the $j$th perverse graded piece and $\operatorname{gr}_{\mathfrak{m}}^{j-i}$ for the corresponding maximal-ideal graded piece. Oblomkov–Yun conjecture. For all $i,j\in\mathbb{Z}$,

$$\operatorname{gr}_j^P\mathrm{H}^{2i}(J_{\frac{m}{n}})=\operatorname{gr}_{\mathfrak{m}}^{j-i}\mathbb{C}[\mathcal{M}_{\frac{m}{n}}](j).$$

This conjecture predicts an equality between the perverse filtration on compactified-Jacobian cohomology and the maximal-ideal filtration on the arc-space coordinate ring; the supplied text reports no resolution.