Beauville decomposition conjecture for compactified Jacobian fibrations

Let $\bar J_{\mathcal C} \xrightarrow{\pi} B$ be a compactified Jacobian fibration associated to a flat family of integral projective locally planar curves of arithmetic genus $g$ over a nonsingular base $B$. Assume that $\bar J_{\mathcal C}$ is nonsingular and that $\pi$ is Lagrangian with respect to a holomorphic symplectic form on $\bar J_{\mathcal C}$. A decomposition

$$\mathfrak h(\bar J_{\mathcal C})=\bigoplus_{i=0}^{2g}\mathfrak h_i(\bar J_{\mathcal C})\in\operatorname{CHM}(B),\qquad \mathfrak h_i(\bar J_{\mathcal C})=(\bar J_{\mathcal C},\bar{\mathfrak p}_i,0)$$

whose homological realization splits the perverse filtration on $R\pi_*\mathbb Q_{\bar J_{\mathcal C}}$ should exist and satisfy stability under Fourier transform,

$$\bar{\mathfrak p}_j\circ\mathfrak F\circ\bar{\mathfrak p}_i=0\qquad\text{for }i+j\neq 2g,$$

multiplicativity with respect to the cup product,

$$\bar{\mathfrak p}_k\circ[\Delta^{\mathrm{sm}}_{\bar J_{\mathcal C}/B}]\circ(\bar{\mathfrak p}_i\times\bar{\mathfrak p}_j)=0,$$

and $c_{2i}(T_{\bar J_{\mathcal C}})\in A^{2i}\mathfrak h_{2i}(\bar J_{\mathcal C})$. This conjectural decomposition would provide a motivic splitting of the perverse filtration compatible with the Fourier transform, the cup product, and the Chern classes; the cited source formulates these properties as Conjectures 3.1 and 3.6.