Isoperimetric conjecture for finite vertex-transitive graphs

Let $G$ be a finite, connected, vertex-transitive graph. Write $\operatorname{diam}(G)$ for its diameter and, for a vertex set $S$, let $\partial S$ denote its edge boundary. Isoperimetric conjecture. If

$$\operatorname{diam}(G)<|G|^{\alpha},$$

then

$$|\partial S|>c_{\alpha}|S|^{1-\alpha}$$

for every $S$ with $1\leq |S|\leq |G|/2$. This would strengthen the known lower bound for the Cheeger constant of a finite transitive graph in terms of the reciprocal of its diameter; the source gives no resolution of the conjecture.