Motivic $\mathfrak{sl}_2$-triple for Lagrangian compactified Jacobian fibrations

Let $\pi:\overline{J}_C\to B$ be as in the motivic Beauville decomposition conjecture, and let $\operatorname{Corr}^k_B(\overline{J}_C,\overline{J}_C)$ denote the degree-$k$ relative correspondences. Motivic $\mathfrak{sl}_2$-triple conjecture. There exists an $\mathfrak{sl}_2$-triple

$$e_0\in\operatorname{Corr}^1_B(\overline{J}_C,\overline{J}_C),\qquad f_0\in\operatorname{Corr}^{-1}_B(\overline{J}_C,\overline{J}_C),\qquad h_0\in\operatorname{Corr}^0_B(\overline{J}_C,\overline{J}_C)$$

which induces the Fourier-stable multiplicative motivic decomposition, with

$$h_0\circ\mathfrak{p}_i=(i-g)\mathfrak{p}_i,\qquad 0\leq i\leq 2g.$$

This conjecturally lifts the generalized Beauville $\mathfrak{sl}_2$-action from cohomology to relative Chow motives and is conditional on the preceding motivic decomposition.