Joyce's nonidentification conjecture for generic mirror discriminants

Let $M$ and $\hat M$ be generic mirror Calabi–Yau $3$-folds. Suppose there exist special Lagrangian fibrations

$$f:M\rightarrow B,\qquad \hat f:\hat M\rightarrow\hat B.$$

Let $\Delta_f\subset B$ and $\Delta_{\hat f}\subset\hat B$ be their discriminants. Joyce's nonidentification conjecture. Even if these fibrations exist, it is not in general possible to homeomorphically identify $B$ and $\hat B$ in a way that identifies $\Delta_f$ and $\Delta_{\hat f}$ and makes the nonsingular fibres of $f$ and $\hat f$ into $3$-tori with dual homology. This challenges a naive global form of the SYZ picture: although mirror fibrations may exist, generic perturbations can produce nonhomeomorphic discriminant loci, so the bases need not be identifiable in the required fibrewise-dual manner.