Multiplicative splitting conjecture for the perverse filtration on compactified Jacobians

Let $C$ be a locally planar integral curve of arithmetic genus $g$, let $\overline{J}_C$ be its compactified Jacobian, and let $P_\bullet H^*(\overline{J}_C,\mathbb{Q})$ be the perverse filtration. A splitting is a decomposition

$$H^*(\overline{J}_C,\mathbb{Q})=\bigoplus_{i=0}^{2g}H^*_{(i)}(\overline{J}_C,\mathbb{Q})$$

with

$$P_kH^*(\overline{J}_C,\mathbb{Q})=\bigoplus_{i\leq k}H^*_{(i)}(\overline{J}_C,\mathbb{Q}).$$

Multiplicative splitting conjecture. There exists such a splitting that is multiplicative under cup product:

$$\cup:H^*_{(i)}(\overline{J}_C,\mathbb{Q})\times H^*_{(i')}(\overline{J}_C,\mathbb{Q})\longrightarrow H^*_{(i+i')}(\overline{J}_C,\mathbb{Q}).$$

The conjecture asks for a multiplicative refinement of the perverse filtration. Prediction A, compatibility of the perverse filtration with cup product, is stated as proven for any locally planar integral curve, while the multiplicative splitting itself remains open; the source also notes that a global family can fail to admit such a splitting.