The Gu–Xu normalized scalar-curvature sphere conjecture

Let $M^n$, with $n\geq4$, be a closed and simply connected Riemannian manifold. Denote by $R_0$ its normalized scalar curvature and by $K_{\max}$ its maximum sectional curvature.

Gu–Xu's conjecture. If

$$R_0 > \frac{3}{5}K_{\max},$$

then $M$ is diffeomorphic to $\mathrm{S}^n$.

The conjecture is motivated by the example of the Cayley projective plane and strengthens an earlier theorem of Gu and Xu with the constant $\frac{12}{5n(n-1)}$. The supplied text does not report a proof or disproof, so the conjecture remains open here.