Finite-CW conjecture for Vietoris–Rips complexes of spheres

Let $S^n$ be the $n$-sphere, let $r\ge 0$, and let $\mathrm{VR}(S^n;r)$ denote the Vietoris–Rips complex formed using the strict-distance convention of the paper. Finite-CW conjecture. The complex $\mathrm{VR}(S^n;r)$ is homotopy equivalent to a finite CW complex for every $r\ge 0$. The analogous assertion is false under the non-strict convention, where certain complexes can be homotopy equivalent to uncountable wedge sums of spheres; the strict-convention statement remains open.