The smooth Cayley fibration non-existence conjecture for full-holonomy Spin(7)-manifolds

A torsion-free ${\mathrm{Spin}}(7)$-manifold is a Riemannian manifold with torsion-free ${\mathrm{Spin}}(7)$-structure, and a smooth Cayley fibration is a smooth fibration whose fibers are Cayley submanifolds. The holonomy is full when it equals ${\mathrm{Spin}}(7)$. Cayley fibration non-existence conjecture. A torsion-free ${\mathrm{Spin}}(7)$-manifold with full holonomy does not admit a smooth Cayley fibration. This is motivated by the expectation that calibrated fibrations on manifolds with full exceptional holonomy must develop singular fibers; the paper gives evidence for the claim and rules out smooth Cayley fibrations on all known examples of compact torsion-free ${\mathrm{Spin}}(7)$-manifolds, but the general statement remains open.