DeVos–Mohar depth expansion conjecture for vertex-transitive graphs

Let $X=(V,E)$ be a connected vertex-transitive graph and let $A\subset V$ be finite, with $0<|A|\leqslant |V|/2$. Define

$$depth(A)=\sup\{d(u,V\smallsetminus A)\mid u\in A\}.$$

DeVos–Mohar conjecture. There exists a fixed constant $c>0$ such that

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

The conjecture proposes replacing the diameter in the Babai–Szegedy isoperimetric inequality by the depth of the set. It is refuted by examples in the Cayley graph of $\mathbb{Z}^2$, and consequently in suitable finite vertex-transitive graphs.