Metric Deformations from Non-Negative to Positive Curvature

As Benoît Kloeckner points out this is false for non simply connected manifolds with $RP^2\times RP^2$ being a counterexample (by Synge's theorem). For simply connected manifolds this is a well known open problem.

BTW, Sha-Yang examples are only known not to admit nonnegative sectional curvature for connected sums of things like $S^n\times S^m$ for very large number of summands (by Gromov's betti number estimate). For, say, connected sum of 3 copies of $S^n\times S^m$ with $n,m>1$ nothing is known.

Also, there are plenty of easier examples of manifolds of positive Ricci curvature other than those of Sha and Yang. For example homogeneous spaces and more generally biquotients of compact Lie groups with finite fundamental groups. All of them have positive Ricci curvature and nonnegative sectional curvature but almost none are known to admit positive sectional curvature.

Moreover, there are lots of such examples with quasi-positive curvature ( where sectional curvature is positive on a dense open set of points in $M$) and the question is still open for such manifolds. See for example this paper by Wilking "Manifolds with positive sectional curvature almost everywhere."

Among many other examples Wilking constructs such metrics on $S^2\times S^3$. This shows that even in the simply connected case either the Hopf conjecture (that a product of positively curved manifolds can not be positively curved) is false or the deformation conjecture for quasi-positively curvaed manifolds is false.

I think the answer is no, because if I remeber well $\mathbb{R}\mathrm{P}^2 \times \mathbb{R}\mathrm{P}^2$ does not admit a positively curved metric. My reference for this is Gallot-Hulin-Lafontaine, but I do not have the book at hand right now.