Benjamini's thick embedding conjecture for transitive graphs

Let $d\geq 3$, and let $Y$ be a locally finite, non-planar, transitive graph with separation profile $\simeq n^\alpha$ for some $\alpha \in (0,1]$. A $1$-thick embedding of a graph into $Y$ is an embedding in which distinct vertices and edges are separated according to the unit-thickness condition. The volume of an embedding is measured using the graph measure, which assigns length to edges. Benjamini's conjecture. There exists a constant $C$ such that for every graph $G$ with $n$ vertices and maximum degree at most $d$, there is a $1$-thick embedding of $G$ into $Y$ with volume at most $Cn^{1/\alpha}$, possibly up to a factor of $\operatorname{polylog}(n)$. The conjecture predicts that the separation profile gives the correct polynomial upper bound for thick embedding volumes of bounded-degree graphs into suitable transitive graphs; the supplied text gives no resolution status.