Asymptotic disintegrability of compact connected Riemannian manifolds

Let $(\mathfrak{X},d,\mu)$ be a metric probability space, where $\mathfrak{X}$ is a compact, connected, $D$-dimensional Riemannian manifold, $d$ is its geodesic distance, and $\mu$ is absolutely continuous with respect to volume measure with density $f$ satisfying

$$0<c\leq f(x)\leq C<\infty$$

for all $x\in\mathfrak{X}$. Riemannian-manifold disintegrability conjecture. For any $\alpha\in(0,\infty)$, $(\mathfrak{X},d,\mu)$ is asymptotically disintegrable with rate

$$\left(O(n^{-\alpha}),O\!\left(\left(\frac{\ln n}{n}\right)^{1/\max\{2,D\}}\right)\right).$$

This would extend the paper's examples of asymptotically disintegrable spaces beyond intervals, circles, graphs, and cubes; the proposed rate is intended to support near-optimal high-probability Wasserstein bounds. The conjecture remains open.