Asymptotic growth conjecture for covering systems over $Z_s^q$

Let $R(s,q)$ denote the minimum size of a covering system of vectors over $Z_s^q$. For fixed $s\geq 3$, consider the growth of $R(s,q)$ as $q$ tends to infinity. Asymptotic growth conjecture. For any fixed $s\geq 3$,

$$R(s,q)=o(2^q),$$

and furthermore

$$R(s,q)=\Theta(2^q/q^{c}),$$

where $c=c(s)>0$ is a constant that depends only on $s$. The preceding discussion gives a lower bound of order $2^q/\sqrt{q}$ when $s=3$, while the available upper bounds do not establish the conjectured asymptotic form; determining the constants $c(s)$ remains open.