  Let $G$ be a compact connected Lie group and let $K$ be a closed subgroup of
$G$. In this paper we study whether the functional $g\mapsto
\lambda_1(G/K,g)\operatorname{diam}(G/K,g)^2$ is bounded among $G$-invariant
metrics $g$ on $G/K$. Eldredge, Gordina, and Saloff-Coste conjectured in 2018
that this assertion holds when $K$ is trivial; the only particular cases known
so far are when $G$ is abelian, $\operatorname{SU}(2)$, and
$\operatorname{SO}(3)$. In this article we prove the existence of the mentioned
upper bound for every compact homogeneous space $G/K$ having multiplicity-free
isotropy representation.
