Lower semicontinuity of the homotopy type of Vietoris-Rips complexes

Let $X$ be a compact metric space, let $r\in\mathbb{R}_{>0}$, and let $\mathrm{VR}_r(X)$ denote the Vietoris-Rips complex at scale $r$.

Lower semicontinuity conjecture. The complex $\mathrm{VR}_r(X)$ is homotopy equivalent to $\mathrm{VR}_{r-\varepsilon}(X)$ whenever $\varepsilon>0$ is sufficiently small.

The paper explicitly labels this conjecture as false and provides a counterexample immediately afterward. Thus the proposed lower semicontinuity statement does not hold for all compact metric spaces.