Isomorphism between the sign-twisted curve complex and the Lagrangian sphere complex of the intersection of two quadrics

Determine whether the simplicial injection from the non-separating subcomplex of the sign-twisted curve complex of the genus-two surface into the Lagrangian sphere complex of the intersection of two generic quadrics in [?]mathbb{CP}^5 is a simplicial isomorphism.

Background

For the genus-two odd character variety $M_2^{\mathrm{odd}}$, identified symplectically with the intersection $Q_1\cap Q_2$ of two generic quadrics in $\mathbb{CP}^5$, the paper constructs a simplicial injection from the subcomplex $\mathcal N(\Sigma_2)$ of non-separating vertices in the sign-twisted curve complex to the Lagrangian sphere complex $\mathcal L(Q_1\cap Q_2)$.

The unresolved issue is whether every Lagrangian sphere in $Q_1\cap Q_2$ arises from a sign-twisted non-separating curve. The paper notes that this would follow from two additional results: an isomorphism for the analogue of the mapping-class-group homomorphism for $M_2^{\mathrm{odd}}$, and transitivity of the symplectomorphism group of $Q_1\cap Q_2$ on Lagrangian spheres up to symplectic isotopy. Neither assertion is established there.