Simultaneous primitive roots: a conditional lower bound for prime bases

For every fixed finite set of distinct positive primes, we prove that at least $cx/(\log x{)}^2$ primes in $(x,2x)$ have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.