Eldredge–Gordina–Saloff-Coste eigenvalue–diameter conjecture

Let $G$ be a compact connected Lie group, and let $\mathcal M^G$ denote the space of left-invariant Riemannian metrics on $G$. For $g\in\mathcal M^G$, write $\lambda_1(G,g)$ for the first positive Laplace eigenvalue and $\operatorname{diam}(G,g)$ for the diameter. Eldredge–Gordina–Saloff-Coste conjecture. There is a positive real number $C$, depending only on $G$, such that

$$\lambda_1(G,g) \leq \frac{C}{\operatorname{diam}(G,g)^2}$$

for every $g\in\mathcal M^G$. This conjecture asserts a uniform upper bound for the product $\lambda_1(G,g)\operatorname{diam}(G,g)^2$ within the family of left-invariant metrics; it is known for uniformly doubling compact connected Lie groups, but the general case remains open.