The logarithmic upper-bound conjecture for the least quadratic nonresidue

Let $p$ be a prime, and let $n_p$ denote the least positive quadratic nonresidue modulo $p$. Here $f(p)\ll g(p)$ means that there is a constant $C>0$ such that $f(p)\leq Cg(p)$ for all sufficiently large $p$. Logarithmic upper-bound conjecture. For every large prime $p\geq 3$,

$$n_p\ll (\log p)(\log \log p).$$

This is a conjectured improvement over the classical bounds for the least quadratic nonresidue, lying between the known polylogarithmic consequences of stronger hypotheses and the much stronger Vinogradov conjecture. Its status is not resolved in the supplied source context.