Exponential concentration implies a Poincaré inequality under non-negative Ricci curvature

Let $(\mathcal{X},Q,\pi)$ be a Markov chain with invariant measure $\pi$, let $\operatorname{Ric}(\mathcal{X},Q,\pi)\geq0$, and suppose that $\pi$ satisfies a concentration property with respect to the distance $d_{\mathcal{W}}$ with profile

$$\alpha(r)=Me^{-\rho r}.$$

Exponential-concentration Poincaré conjecture. There exists a constant $C(M)$ such that

$$\operatorname{PI}\bigl(C(M)\rho^{-2}\bigr)$$

holds. This is proposed as a discrete analogue of corresponding results in continuous spaces; the conjectured constant depends only on $M$, not on further features of the chain.