The strict-growth conjecture for equivariant covering numbers

Let $G$ be a finite group, and let $\operatorname{cov}_G(d)$ denote the $G$-covering number of a classifying space $E_dG$.

Strict-growth conjecture. For all $d\geq 0$,

$$\operatorname{cov}_G(d)<\operatorname{cov}_G(d+1).$$

This is proposed as a weaker alternative to the generalized Borsuk covering-number conjecture. The paper leaves it open and asks for proofs or counterexamples, together with computations for small groups and low dimensions.