Artin conjectured that if $a$ is an integer which is not a square and not $-1$ then $a$ is a primitive root for infinitely many primes. This conjecture has not been resolved, but partial results are known: Heath-Brown showed that there are at most two prime numbers $a$ for which the conjecture fails.

I'd like to know if a different kind of partial result is known. Let $I(p)$ denote the index of the subgroup of $(\mathbf{Z}/p\mathbf{Z})^{\times}$ generated by 2. Thus $I(p)=1$ if and only if 2 is a primitive root mod $p$. Can one show that there is an infinite sequence of primes in which $I$ remains bounded?