Subgraph-closed classes with strongly sublinear separators have subexponential expansion

Let ${\cal G}$ be a subgraph-closed class of graphs. Suppose that ${\cal G}$ has strongly sublinear separators. Subexponential expansion conjecture. The expansion of ${\cal G}$ is subexponential; that is, there exists a function $f$ with $\log f(k)=o(k)$ such that the expansion of ${\cal G}$ is bounded by $f$. This would remove the bounded-maximum-degree hypothesis from the paper's theorem and give an almost precise characterization of graph classes with sublinear separators in terms of subexponential expansion.