  Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given
compact connected Lie group $G$, there is a positive real number $C$ such that
$\lambda_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all left-invariant metrics
$g$ on $G$. In this short note, we establish the conjecture for the small
subclass of naturally reductive left-invariant metrics on a compact simple Lie
group.
