TITLE: Is it known whether a closed simply-connected manifold of non-negative curvature admits positive Ricci?
QUESTION [6 upvotes]: It is discussed in this question whether a simply-connected closed Riemannian manifold with non-negative Ricci curvature admits positive Ricci curvature, and the answer appears to be "no, there are counter-examples and known obstructions".
My question is this: are there any known examples of closed, simply-connected manifolds of non-negative sectional curvature that do not admit positive Ricci curvature? Are there known obstructions?

REPLY [3 votes]: No. Every simply connected compact manifold of nonnegative sectional curvature admits positive Ricci curvature. This has been proven by Boehm--Wilking: https://link.springer.com/article/10.1007/s00039-007-0617-8
Theorem A. Let $(M^n,g)$ be a compact Riemannian manifold with finite fundamental group and nonnegative sectional curvature. Then $M^n$ admits a metric with positive Ricci curvature.