Monotonicity conjecture for the connectivity of Čech complexes of spheres

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Write $\operatorname{conn}(K)$ for the connectivity of a complex $K$. Connectivity monotonicity conjecture. The function

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

is non-decreasing on $(0,\pi)$. This is known for $n=1$, and for $n\ge 2$ the complexes are known to be simply connected; for $n\ge 3$, it remains unknown whether the connectivity stays at least $n-1$ once $r\ge \pi/2$.