There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally
an odd prime power, so that $(\mathbb Z/q \mathbb Z)^\ast$ is cyclic, and has generators, called *primitive root*). I wonder if there is any conjecture or result on the following problem:

For $q$ a prime or an odd prime power, estimate the smallest prime $n(q)$ which is

nota primitive root modulo $q$.

It is certainly very hard to prove any non-trivial lower bound on $n(q)$, since the statement *$n(q)>2$ for all prime $q$ sufficiently large* is clearly equivalent to the falsity of Artin's conjecture
that $2$ is a primitive root infinitely often. But what about a conjectural lower bound?
and what about upper bound?