If Artin's conjecture on primitive roots is true, then 2 generates $\mathbb{Z}_p^*=\{1,2,\ldots, p-1\}$ for infinitely many primes $p$. My question is that can one at least show that $\mathbb{Z}_p^*$ is generated by 2 and 3 for infinitely many primes $p$?

More clearly, is it true that for infinitely many primes $p$ the following is correct: every natural number $m$ not divisible by $p$ is congruent to $2^a3^b$ mod $p$ for some natural numbers $a,b$?