The cyclic-group Borsuk covering conjecture

For $m\geq 2$ and $d\geq 0$, let $\operatorname{cov}_{\mathbb{Z}_m}(d)$ denote the $\mathbb{Z}_m$-covering number of a classifying space $E_d\mathbb{Z}_m$. The general lower bound is $m+d\leq \operatorname{cov}_{\mathbb{Z}_m}(d)$.

Cyclic-group Borsuk covering conjecture.

$$\operatorname{cov}_{\mathbb{Z}_m}(d)=m+d.$$

This is a less general conjecture that may remain true even if the generalized Borsuk covering-number conjecture fails. The paper states that both conjectures are open and points to computational work in dimensions $2$ and $3$.