$\DeclareMathOperator{\ord}{ord}$Artin's conjecture stipulates that $\ord_p(2) = p -1$ for infinitely many primes $p$, where $\ord_p(2)$ denotes the multiplicative order of $2$ modulo $p$. More generally one expects that $\ord_p(2)$ is often quite large. I'm looking for a weakened version of this, namely:
Does the sum $\displaystyle\sum_{p \leq x} \frac{1}{\ord_p(2)^2}$ converge as $x\to\infty$?
I would prefer unconditional results, but results conditional on e.g. GRH are still welcome.
