Homological cop number conjecture for simplicial pseudomanifolds

Suppose that $X$ is an $n$-dimensional simplicial pseudomanifold, meaning that each simplex is contained in an $n$-simplex and each $(n-1)$-simplex is contained in at most two $n$-simplices. Let $H_i(X)$ have rank $r_i$ for $i=1,\dots,n$. Homological cop number conjecture.

$$c(X)=O\left(n\sqrt{r_1+\cdots+r_n}\right).$$

This proposes a bound on the cop number in terms of the dimension and the ranks of the positive-degree homology groups. The paper presents it as a basic open question about which geometric properties force large cop number.