Primes with small primitive roots

Let $\delta (p)$ tend to zero arbitrarily slowly as $p\to \mathrm{\infty }$. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le p^{1\mathrm{/}4-\delta (p)}$ for all $p\in \mathcal{S}$, where $\mathrm{\#}\{p\le x:p\in \mathcal{S}\}\sim \pi (x)$ as $x\to \mathrm{\infty }$.