The least square-free primitive root bound beyond 409

Let $g^{\square}(p)$ denote the least square-free primitive root modulo a prime $p$. Least square-free primitive root conjecture. For all $p>409$ we have

$$g^{\square}(p)<\sqrt{p}-2.$$

The preceding explicit bound proves only that $g^{\square}(p)<p^{0.96}$ for every prime $p$; this conjecture proposes a substantially stronger square-root-scale bound, motivated by computational evidence and related work.