Duality conjecture for the genus-five Prym fibration

Let $S'$ be a K3 surface that is a branched double cover of a degree-one del Pezzo surface $T'$, and let $D'$ be a genus-two curve on $T'$ covered by a genus-five curve $C'$ on $S'$. Denote by

$$\overline{\operatorname{Prym}}(\mathcal{C}'/\mathcal{D}')$$

the resulting six-dimensional holomorphic symplectic orbifold with its Lagrangian fibration over $\mathbb{P}^3$. Duality conjecture for the genus-five Prym fibration. The dual fibration of $\overline{\operatorname{Prym}}(\mathcal{C}'/\mathcal{D}')$ is a special Matteini system, or possibly a degeneration of Matteini systems. The parameter-count discussion explains why the duality may involve a special member or degeneration rather than a generic Matteini system.