Are there infinitely many primes $n$ such that $\mathbb{Z}_n^*$ is generated by $\{ -1,2 \}$?

Let $n$ a prime, and let $\mathbb{Z}_n$ denote the integers modulo $n$. Let $\mathbb{Z}^*_n$ denote the multiplicative group of $\mathbb{Z}_n$

Are there infinitely many $n$ such that $\mathbb{Z}^*_n$ is generated by $\{ -1, 2 \}$?

Artin's conjecture on primitive roots implies something even stronger: that there are infinitely many $n$ such that $\mathbb{Z}^*_n$ is generated by $\{ 2 \}$. Although likely to be true (in particular it is implied by the Generalized Riemann Hypothesis), as far as I know this conjecture remains open. I am wondering if it is possible that with generators $\{-1,2 \}$, this is known unconditionally.

(One could, of course, ask this for any two generators. For reasons that I'll omit here, I am especially interested in the the generating set $\{-1,2 \}$.)

Solution 1:

I am afraid this is out of reach. As quid comments, one can do this with three prime generators, but even two prime generators is too hard; and including $-1$ as a generator does not help much.