The automaticity of the set of primes

For an integer $q\ge 2$, let $A=\{0,1,\dots ,q-1\}$. The q-automaticity $A\left(x\right)$ of a set $X$ is the size of the smallest automaton over the alphabet $A$ that recognizes $X$ on all words of length ≤x. We show that the q-automaticity of the set of primes is at least $x\exp (-c{(\log \log x)}^2\log \log \log x)$, which is fairly close to the maximal q-automaticity.