Classification conjecture for Lagrangian fibrations by Jacobians

Let $\mathcal{C}\rightarrow\mathbb{P}^n$ be a family of reduced and irreducible curves of arithmetic genus $n$. Suppose that the relative compactified Jacobian

$$X:=\overline{\operatorname{Jac}}^d(\mathcal{C}/\mathbb{P}^n)$$

is an irreducible holomorphic symplectic manifold, and hence a Lagrangian fibration via the support map to $\mathbb{P}^n$. Classification conjecture for Lagrangian fibrations by Jacobians. The fibration $X\rightarrow\mathbb{P}^n$ is a Beauville–Mukai system: the family $\mathcal{C}\rightarrow\mathbb{P}^n$ is a complete linear system of curves on a K3 surface. The conjecture asserts that the Beauville–Mukai integrable system is the only compact Lagrangian fibration by Jacobians under these hypotheses.