Finite CW-complex conjecture for Čech complexes of spheres

For each dimension $n\ge 1$ and scale $r>0$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r$. Finite CW-complex conjecture. The complex

$$\mathrm{\check{C}}(S^n;r)$$

is homotopy equivalent to a finite CW complex. The claim concerns the finiteness of the homotopy type despite the generally infinite vertex set of the Čech complex; no resolution is given in the source.