Motivated by the Chinese Remainder Theorem, I'm interested in common primitive roots modulo several primes.

Let's first look at common primtive roots modulo two distinct primes. For any distinct primes $p$ and $q$, does there exist a positive integer $g\leqslant \sqrt{4pq+1}$ which is a primitive root modulo $p$ and also a primitive root modulo $q$? In August 2017, I conjectured that the answer is yes. Moreover, I think that we may even replace $\sqrt{4pq+1}$ by $\sqrt{pq}$ if $\{p,q\}$ is not among the following 15 pairs of primes \begin{gather*}\{2,3\},\ \{2,11\},\ \{2,13\},\ \{2,59\},\ \{2,131\}, \ \{2,181\}, \ \{3,7\},\ \{3,31\}, \\\{3,79\},\ \{3,191\},\ \{3,199\},\ \{5,271\}, \ \{7,11\},\ \{7,13\}, \ \{7,71\}. \end{gather*}

In general, for each integer $n>1$, I guess that there is a positve constant $c_n$ such that for any $n$ distinct primes $p_1,\ldots,p_n$ there is a positive integer $g \leqslant c_n\root{n}\of{p_1\cdots p_n}$ which is a primitive root modulo $p_k$ for all $k = 1,\ldots,n$.

See http://oeis.org/A291690 for related data and comments.

Any ideas towards my question on common primitive roots modulo several primes?