The metric SYZ conjecture

Let $(X_t,\omega_{CY,t})$ be a polarised degeneration family of $n$-dimensional Calabi–Yau manifolds near the large complex structure limit, and let $\mu_t$ denote the normalised Calabi–Yau measure. Metric SYZ conjecture. There exist special Lagrangian $T^n$-fibrations on some subset $U_t\subset X_t$ for $0<|t|\ll 1$ such that

$$\mu_t(U_t)\to 1$$

as $t\to 0$. This is the metric version of the SYZ conjecture and is presented as a major motivation for the paper; the source describes the problem as unresolved and discusses the expected semi-flat metric asymptote in the generic region.