Conjecture on the least stationary primitive root modulo p

Let $p$ be a prime, and let $g_s(p)$ denote the least stationary primitive root in $\mathbb{Z}/p\mathbb{Z}$. Let $\varepsilon>0$ be a small number. Least stationary primitive root conjecture. The least stationary primitive root satisfies

$$g_s(p)\ll (\log p)^{1+\varepsilon},$$

as $x\to\infty$. The preceding result gives the weaker bound $g_s(p)\ll p^{1/2+\varepsilon}\log p$, while the least stationary primitive root is bounded above by the least primitive root modulo $p^2$; the conjectured logarithmic bound would substantially improve the available estimates.