Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g_1$ satisfying $\mathrm{sec}(M,g_1) > 0$?

I understand that this question is ludicrous, at best (for instance, an affirmative would prove that $S^n\times S^m$ always carries a metric of positive sectional curvature). I'm assuming that there is a counterexample, but I can't seem to figure it out. Most examples of manifolds with positive Ricci curvature that I know can't admit metric of non-negative sectional curvature for topological reasons, for instance the Sha--Yang construction of metrics with positive Ricci curvature on connected sums of $S^n \times S^m$.

I was wondering about this in view of the following two results: The first, due to Aubin and Ehrlich, asserts that a metric $g$ with $\mathrm{Ricci}(M,g) \geq 0$ everywhere and $\mathrm{Ricci}(M,g) > 0$ somewhere can be deformed into a metric $g_1$ with $\mathrm{Ricci}(M,g_1) > 0$ everywhere. The second, due to Gao--Yau, asserts that if $\mathrm{Ricci}(M,g) > 0$ everywhere, then $g$ can be deformed into a metric $g_1$ with $\mathrm{Ricci}(M,g) > 0$ everywhere and $\mathrm{sec}(M,g) > 0$ somewhere.

The Gao--Yau result is a local solution to the intial question.

As a related question: Suppose the Ricci curvature of $(M,g)$ is pinched. Does this change the answer to the question if the pinching constant is particulary small or would it seem that pinching plays no role in this? Is there any theory of pinched Ricci curvature?

T. Aubin, Metriques riemannienes et courbure. Diff. Geom., 4:383--424, 1970

P. Ehrlich, Metric deformations of curvature. Geom. Ded., 5:1--23, 1976

L. Gao, S.-T. Yau, The existence of negatively Ricci curved metrics on three manifolds. Invent. Math., 85:637--652, 1986