Metric Latschev theorem for infinite metric spaces

Let $M$ be a closed Riemannian manifold, and let $d_{\mathrm{GH}}$ denote the Gromov–Hausdorff distance. For sufficiently small $r$, the metric Vietoris–Rips thickening $\mathrm{VR}^m_<(X;r)$ is defined for any metric space $X$. Metric Latschev conjecture. For any $r$ sufficiently small, there exists a $\delta$ such that every, possibly infinite, metric space $X$ satisfying

$$d_{\mathrm{GH}}(M,X)<\delta$$

also satisfies

$$\mathrm{VR}^m_<(X;r)\simeq M.$$

This extends the finite-metric-space analogue of Latschev’s theorem; the extension to infinite $X$ is stated to be currently unknown.