Wilhelm's conjecture on the fiber dimension of positively curved submersions

Let $\pi\colon (M^{n+k},g)\to (B^n,b)$ be a Riemannian submersion. If $(M,g)$ is compact and has positive sectional curvature, then $k<n$.

Wilhelm's conjecture. Every such submersion satisfies

$$k<n.$$

This is the conclusion of Wilhelm's conjecture for compact positively curved total spaces; the source notes that González and Radeschi proved it for submersions from spaces homotopically equivalent to known examples with positive sectional curvature, without curvature assumptions. The general statement is presented here without evidence of resolution.