Abrishami–Czyżewska–Kluk–Pilipczuk–Pilipczuk–Rzążewski coarse separator conjecture

Let $G$ be a graph. For $r\in\mathbb{N}$, an $r$-cover of a set $S\subseteq V(G)$ is a set $\hat{S}\subseteq V(G)$ such that $S\subseteq N^r_G[\hat{S}]$. A set is $(k,r)$-coverable if it has an $r$-cover of size at most $k$. The graph $G$ admits $(k,r)$-balanced separators if, for every weight function $w:V(G)\to\mathbb{R}_{\geq 0}$, it has a $(k,r)$-coverable $w$-balanced separator. A tree decomposition is $(k',r')$-coverable if every bag is $(k',r')$-coverable.

Abrishami–Czyżewska–Kluk–Pilipczuk–Pilipczuk–Rzążewski's conjecture. For every $k,r\in\mathbb{N}$, there exist $k',r'\in\mathbb{N}$ such that if $G$ admits $(k,r)$-balanced separators, then $G$ admits a $(k',r')$-coverable tree decomposition.

This is proposed as a coarse analogue of the linear correspondence between treewidth and separation number. The converse implication, from a tree decomposition with $(k,r)$-coverable bags to $(k,r)$-balanced separators, follows by a standard argument; the conjectured direction would therefore preserve this correspondence under quasi-isometry.