Extension of the empirical Wasserstein barycenter rate to the regime $1\leq\beta-\alpha<2$

Let $P$ be the probability measure and let $\bar P$ be its Wasserstein barycenter. Let ${\bar P}_n$ denote the empirical Wasserstein barycenter based on $n$ observations, and let $\alpha$ and $\beta$ be the strong-convexity and smoothness parameters from the setting of the barycenter convergence theorem. Empirical barycenter rate conjecture. With the notation of the barycenter convergence theorem,

$$\mathbb{E}\left[\mathcal{W}^2({\bar P}_n,\bar P)\right] \leq \frac{C}{n}\,\frac{\mathbb{E}_{\mu\sim P}[\mathcal{W}^2(\mu,\bar P)]}{(\beta-\alpha-2)^2},$$

for all $1\leq\beta-\alpha<2$, with a constant $C>0$ that does not depend on $P$. The conjecture would extend the existing analysis beyond its stated range; the paper notes that the sharpness of the available support bounds on spheres leaves this extension open.