Chow-theoretic perversity conjecture for Chern classes of compactified Jacobian fibrations

Let $\pi:\overline{J}_C\to B$ be a Lagrangian compactified Jacobian fibration as in the motivic Beauville decomposition conjecture, with motivic summands $h_{2i}(\overline{J}_C/B)$. Chow-theoretic Chern-class conjecture. For every $i\geq 0$,

$$c_{2i}(\overline{J}_C)\in\operatorname{CH}^{2i}(h_{2i}(\overline{J}_C/B),\mathbb{Q}).$$

This prediction refines the Beauville–Voisin perspective by placing the even Chern classes in the expected Chow-theoretic perversity components for Lagrangian compactified Jacobian fibrations over possibly non-proper bases.