The $s=3$ covering-system asymptotic conjecture

Let $R(s,q)$ denote the minimum size of a covering system of vectors over $Z_s^q$. In particular, consider the case $s=3$ as $q$ tends to infinity. The $s=3$ asymptotic conjecture.

$$R(3,q)=\Theta(2^q/\sqrt{q}).$$

The source gives a lower bound of order $2^q/\sqrt{q}$ for $R(3,q)$ and notes that this conjecture would make that lower bound essentially optimal; proving the matching upper bound remains open.