Yau's pinching problem for sectional and normalized scalar curvature

Let $(M^{n},g)$ be a closed, simply connected Riemannian manifold of dimension $n\ge 4$. Define the minimum sectional curvature and normalized scalar curvature at $x\in M$ by

$$K_{\min}(x)=\min_{\sigma\subset T_xM}K(\sigma),\qquad S_0(x)=\frac{\operatorname{Scal}(M^n)|_x}{n(n-1)}.$$

Here $S_0$ is the average of the sectional curvatures at a point. Yau's pinching problem. If

$$K_{\min}>\frac{n-1}{n+2}S_0,$$

pointwise, then $M^n$ is homeomorphic to the $n$-sphere $\mathbb{S}^n$. This is the proposed sectional-scalar curvature analogue of the sphere theorem; the supplied material does not establish whether the sharp formulation is resolved.