The Spin(7) SYZ fibration conjecture

Let $(X,\Phi)$ be a $\operatorname{Spin}(7)$-manifold, and let a toric Cayley mean a Cayley four-fold of the toric type intended in the source. The Spin(7) SYZ fibration conjecture. There exists a compact Riemannian four-manifold $S$ and a singular fibration

$$\pi:X\to S$$

by toric Cayleys. Moreover, the mirror-dual $\operatorname{Spin}(7)$-manifold is

$$\overline{X}=\operatorname{Hom}(\pi_1(\pi^{-1}),\operatorname{U}(1)).$$

This is the proposed extension of the Strominger–Yau–Zaslow picture from Calabi–Yau threefolds to special-holonomy spaces; the source does not establish it.