Expansion vs Decreasing $d$-neighborhood min-cut

Question

Suppose we're given a graph $G$, in which know for (almost) any node-pair, their $d$-neighborhood min-cut (i.e. min-cut defined on their $d$-neighborhood subgraph) is decreasing with $d$. Is there any way to prove that this graph has a low expansion? (or find upper/lower bounds for the expansion)