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Take the standard metric on $\mathbb{CP}^n$ and perturb it in a small neighbourhood so that it is no longer isometric to the original. If the second derivatives of the additional term are small, the metric remains positively curved.
Did you try to contact the author? As a side note, I remember myself looking at this preprint back in 2014 and finding some unfixable flaw in it. The flaw was in something similar to you question. Maybe it is the same place. Also note that the main theorem (Theorem 2) is obviously false: there are positively curved metrics on $\mathbb{CP}^n$ such that it is not a locally symmetric space (just perturb the standard metric to obtain an example).
Yes, as well as for the cylinder over any planar convex set. I was thinking about the strictly convex case. I can prove the local version for smooth strictly convex bodies and $k=2$.
