The generalized Borsuk covering-number conjecture

Let $G$ be a finite group. For a classifying space $E_dG$, write $\operatorname{cov}_G(d)$ for its $G$-covering number, namely the minimum number of sets in a $G$-cover. The preceding bounds give

$$d+|G|\leq \operatorname{cov}_G(d)\leq d(|G|-1)+2.$$

Generalized Borsuk covering-number conjecture.

$$\operatorname{cov}_G(d)=|G|+d.$$

This would make the lower bound sharp for every finite group and every dimension. The paper reports computational evidence, but states that the conjecture remains open; its special cases and the strict-growth assertion below are proposed as related follow-up questions.