Gaussian concentration implies modified logarithmic Sobolev inequalities under Ricci curvature bounds

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

$$\alpha(r)=Me^{-\rho r^2}.$$

Assume that $\operatorname{Ric}(\mathcal{X},Q,\pi)\geq-\kappa$ for some $\kappa>0$. Gaussian-concentration modified logarithmic Sobolev conjecture. There exists a constant $\tau(M)$ and a constant $\lambda(\kappa,M,\rho)$ such that, if

$$\frac{\kappa}{\rho}<\tau(M),$$

then $\operatorname{MLSI}\bigl(\lambda(\kappa,M,\rho)\bigr)$ holds. If moreover $\operatorname{Ric}(\mathcal{X},Q,\pi)\geq0$, then $\operatorname{MLSI}(cM\rho)$ holds for some universal constant $c$. The open aspect identified in the source is the value of the constant.