asymptotic behavior of Lipschitz constants of sectional curvature

I'm studying the paper "Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds" (https://arxiv.org/pdf/1402.4947v1.pdf) and I have some problem in understanding the proof of theorem 2.

The author defines the class $\mathcal{M}_n=\{$ $M$ compact riemannian manifold such that $dim(M)=n$ and $K\geq$ $5d \over 6$$\}$ where $K$ is the sectional curvature of $M$ and $d$ is a positive constant, the diameter of the Grassmannian $G(2,n)$, and a sequence $k_n=\sup_{M_n\in \mathcal{M}_n} (\sup _{m \in M_n} ||K_m||_{lip})$ where $||K_m||_{lip}$ is the Lipschitz constant of the function $K_m: G(2,n)\rightarrow \mathbb{R}$, the sectional curvature at point $m\in M$.
The author shows that $k_n$ is finite for every $n$ and then it seems that he deduces directly that $k_n \rightarrow 0$ thanks to a concentration theorem on Grassmanian (theorem 9). I don't understand how we can apply this theorem to deduce the limit of the sequence. Do you have any ideas?

• 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). – Sergei Ivanov Jul 2 '16 at 20:24
• @SergeiIvanov Thank you for the comment. I have tried to contact the author but I have not received any answer yet. Could you please give me some further information about the counterexample of theorem 2? – Nicolò De Ponti Jul 3 '16 at 17:35
• 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. – Sergei Ivanov Jul 3 '16 at 20:04