Limit inferior of least primitive root is finite

Statement

There is a finite number (at most $5$) that equals the limit inferior of the sequence of least primitive roots modulo prime numbers.

Facts used

1. Heath-Brown theorem on Artin's conjecture

Proof

By the Heath-Brown theorem on Artin's conjecture, there are at most two exceptional positive values of $a$ that are not perfect squares and are primitive roots for only finitely many primes. Thus, in the set $\{2,3,5\}$ there is at least one value that is a primitive root for infinitely many primes. The limit inferior of the least primitive root is bounded by that element, and hence by $5$.