Vietoris–Rips thickening self-reconstruction conjecture

Let $X$ be a metric space, potentially with additional assumptions, let $r\geq 0$, and let

$$M=\operatorname{VR}^{\mathrm{m}}(X;r).$$

Here $\operatorname{VR}^{\mathrm{m}}(X;r)$ denotes the metric Vietoris–Rips thickening at scale $r$. Vietoris–Rips thickening self-reconstruction conjecture. There exists $\varepsilon>0$ such that

$$\operatorname{VR}^{\mathrm{m}}(M;\varepsilon)\simeq M.$$

The conjecture proposes that a Vietoris–Rips metric thickening can be recovered, up to homotopy, by applying a sufficiently small-scale Vietoris–Rips thickening to itself. The source explicitly leaves open what additional assumptions on $X$ may be needed and gives no resolution.