Petersen–Wilhelm fiber dimension conjecture

Let $F\hookrightarrow M\rightarrow B$ be a Riemannian submersion from a positively curved closed manifold $M$. Petersen–Wilhelm's fiber dimension conjecture. Then

$$\dim F<\dim B.$$

This conjecture gives a dimension restriction on the fibers of positively curved Riemannian submersions. The paper proves a stronger version for principal bundles and verifies it for additional classes of submersions, but the general conjecture remains open.