Steinhaus–Čech equivalence conjecture in Euclidean space

Let $X$ be a finite data set in $\mathbb{R}^n$ and let $R>\operatorname{diam}(X)$. Let $\check{C}_R(X)$ denote the cover of $X$ by balls of radius $R$ centered at the points of $X$. Equip the cover with Lebesgue measure.

Steinhaus–Čech equivalence conjecture. The Čech filtration constructed from $X$ is isomorphic to the cover filtration on $X$ constructed from $\check{C}_R(X)$.

This is the paper's more precise Euclidean formulation of the proposed equivalence. The authors prove the corresponding result when $n=1$ and provide experimental evidence and a one-directional result in higher dimensions; the general statement remains open.