Petersen–Wilhelm fiber dimension conjecture for Riemannian submersions

Let $\pi\colon (M^{n+k},\textsl g)\to (B^n,b)$ be a Riemannian submersion. Suppose that $(M,\textsl g)$ is compact and has positive sectional curvature. Petersen–Wilhelm's conjecture. Then

$$k<n.$$

This is the dimension restriction on fibers of positively curved Riemannian submersions, expressed with $\dim M=n+k$ and $\dim B=n$. The source presents this as a restatement of the same Petersen–Wilhelm conjecture and discusses results showing that fatness of the horizontal distribution resolves it in appropriate settings; the general statement remains open.