Depth-bound conjecture for vertex-transitive graphs

Let $G$ be a connected vertex-transitive graph, let $A\subseteq V(G)$ be finite with $0<|A|\leq \tfrac12|V(G)|$, let $\partial A$ denote the vertex boundary of $A$, and let $\operatorname{depth}(A)$ denote the depth of $A$. Depth-bound conjecture. There exists a fixed constant $c>0$ such that

$$\frac{|\partial A|}{|A|}\geq \frac{c}{\operatorname{depth}(A)}.$$

This conjecture would replace the diameter in the Babai–Szegedy boundary estimate by a constant multiple of the depth, and would yield the expected quadratic bound in the paper's main theorem. Its status is not resolved in the supplied source.