mg.metric geometry – Do Compact Universal Covers have Concentration of Measure phenomenon?

I have a sequence of compact Riemannian manifolds $M_n$ with $diam (M_n) to 0$ and finite fundamental groups $pi_1 (M_n)$ so that their universal covers $L_n$ are compact. Suppose $diam ( L_n ) = 1$ for all $n$, and their volumes are normalized so that $ vol (L_n)=1$.

Is it true that the family of metric measure spaces $L_n$ satisfies some concentration of measure phenomenon? Are they a Levy family? This is closely related to this other question: Diameter of universal cover .