The SYZ integrable-system and B-field mirror-symmetry conjecture

Let $\mathcal M$ be the Calabi–Yau moduli space, let $\mathcal M_{\mathbb R}\subset\mathcal M$ be the locus where the B-fields vanish, and let $X,X'$ be a mirror pair. For B-fields $\alpha\in H^2(X,\mathbb R/\mathbb Z)$ and $\beta\in H^2(X',\mathbb R/\mathbb Z)$, write $X_\beta$ and $X'_\alpha$ for the corresponding deformations, and $T_\beta(\alpha)$ and $T_\alpha(\beta)$ for the transported B-fields.

SYZ and integrable-system conjecture. Near the large-complex-structure/large-volume limit: the SYZ picture holds on $\mathcal M_{\mathbb R}$; for a B-field $B'$ on $X'$, $X_{B'}$ admits a special Lagrangian fibration, generally without a section, whose Jacobian is the original special Lagrangian fibration with section on $X$; and, for every pair $\alpha,\beta$, the mirror of $(X_\beta,T_\beta(\alpha))$ is $(X'_\alpha,T_\alpha(\beta))$.

This extends the SYZ picture away from the zero-B-field locus and predicts that mirror symmetry preserves the integrable-system structure while exchanging the relevant fibration and B-field data. The source presents these assertions as an emerging conjectural picture.