Let $M$ be a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$. Is $M$ diffeomorphic to a spherical space form?

I know, by a theorem of Brendle and Schoen, if $M$ is a compact Riemannian manifold of dimension $n>3$ with pointwise $1/4$-pinched sectional curvature, then $M$ is diffeomorphic to a spherical space form. Therefore, I may ask this question: does a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$ admit a metric with pointwise $1/4$-pinched sectional curvature?