For $\kappa \in (0,1)$, let $\lambda(\kappa)$ be the density of primes $p$ having $\mathrm{ord}^{\times}_p{2} < \kappa p$.
Does $\alpha := \liminf_{\kappa \to 0} \lambda(\kappa)/\kappa > 0$? Is it finite?
[F. Pappalardi has shown (J. Number Theory 57, 1996) that conditionally on GRH, for any increasing $\psi : \mathbb{R}^{> 0} \to \mathbb{R}^{> 0}$ going to $+\infty$, it holds $\mathrm{ord}^{\times}_p{2} < p/\psi(p)$ for all but a fraction $\leq O \Big(\frac{\log{\psi(X)}}{\psi(X)} \Big)$ of the primes up to $X$. This does not answer neither question, however.]
