Cheap balanced separator conjecture for graphs with polynomial expansion

Let $G$ be a graph and let $p$ be a polynomial such that the expansion of $G$ is bounded by $p$. A cost assignment is a function

$$\rho:V(G)\to\mathbb{R}_0^+.$$

For integers $t\geq 1$ and $q\geq 0$, a set $C\subseteq V(G)$ is $(\rho/t)$-cheap with $q$ outliers if there is a set $C'\subseteq C$ with $|C'|\leq q$ and

$$\rho(C\setminus C')\leq \rho(G)/t.$$

Here, a balanced separator is a vertex set whose deletion leaves components satisfying the balance condition used in the paper. Cheap balanced separator conjecture. For every polynomial $p$, there exists a function $q:\mathbb{N}\to\mathbb{N}$ such that, for every graph $G$ with expansion bounded by $p$ and every cost assignment $\rho$, each integer $t\geq 1$ admits a balanced separator that is $(\rho/t)$-cheap with $q(t)$ outliers. This is a weakening of the conjecture that graphs with polynomial expansion are fractionally treewidth-fragile; the conclusion is therefore known for every class known to be fractionally treewidth-fragile, but the stated conjecture is not resolved in general.