Fixed-radius coarse separator strengthening

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.

Fixed-radius coarse separator strengthening. For every $k,r\in\mathbb{N}$, there exists $k'\in\mathbb{N}$ such that if $G$ admits $(k,r)$-balanced separators, then $G$ admits a $(k',r)$-coverable tree decomposition.

This strengthens the preceding conjecture by requiring the radius of the tree-decomposition bags to remain exactly $r$, rather than allowing a new radius $r'$. The source presents it as an additional statement of interest; no resolution is given.