Poisson and symplectic-leaf conjecture for the relative compactified Jacobian

Let $\mathcal{C}/\mathcal{B}'$ be the family of curves and let $\mathcal{J}(\mathcal{C}/\mathcal{B}')$ be its relative compactified Jacobian. Let $h$ denote the projection to $\mathcal{B}'$, and let $\pi_{\mathcal{B},\infty}:\mathcal{B}'\to\mathbb{A}^{p+q-1}$ be the map defined in the source. Poisson and symplectic-leaf conjecture. There is a canonical Poisson structure on $\mathcal{J}(\mathcal{C}/\mathcal{B}')$ such that $h$ is an algebraically completely integrable system. Moreover, the generic fibers of

$$\pi_{\infty}:\mathcal{J}(\mathcal{C}/\mathcal{B}')\to\mathcal{B}'\xrightarrow{\pi_{\mathcal{B},\infty}}\mathbb{A}^{p+q-1}$$

are smooth and are the symplectic leaves of this Poisson structure. The source presents this as a conjectural description, with no resolution supplied.