Granville's least-prime primitive-root conjecture

Let $q$ be a fixed integer with $q\ne 1$ and $q\ne n^2$, and let $p$ be a prime. A number $q$ is a primitive root modulo $p$ when its residue class generates the multiplicative group modulo $p$. Granville's least-prime primitive-root conjecture. There exists a prime $p$ such that $q$ is a primitive root modulo $p$ and

$$p\ll (\log q)(\log \log q)^3.$$

This gives a conjectural logarithmic upper bound for the least prime admitting a prescribed nonsquare integer as a primitive root; the source notes that its result provides only a close approximation to this bound.