Vinogradov's conjecture on the least primitive root

Let $p$ be a prime, and let $g(p)$ denote the least primitive root modulo $p$, equivalently the smallest generator of the multiplicative group of integers modulo $p$.

Vinogradov's conjecture. For every $\epsilon>0$, for all sufficiently large primes $p$, one has

$$g(p)\ll p^{\epsilon}.$$

This conjecture predicts that the least primitive root grows more slowly than every positive power of $p$. The supplied text does not state whether it has been resolved.