  In this paper we examine Grosswald's conjecture on $g(p)$, the least
primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH),
and building on previous work by Cohen, Oliveira e Silva and Trudgian, we
resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all
$p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$
for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo
$p$.
