Gartland's breakability conjecture for wall-subdivision-free graphs

For a positive integer $t$, let $\mathcal{L}_t$ denote the graph class used in the source. A graph is $k$-breakable if, for every weight function $w\colon V(G)\to[0,1]$, it has a $w$-balanced separator with a core of size strictly less than $k$. A $w$-balanced separator is a set whose removal leaves every component with weight at most $1/2$, and a core for a set $Y$ is a set $X$ such that $Y\subseteq N[X]$. A subdivision of the $t\times t$-wall is obtained by subdividing edges of that wall.

Gartland's conjecture. For every positive integer $t$, there is an integer $d=d(t)$ such that every $\mathcal{L}_t$-free graph $G$ with no induced subgraph isomorphic to a subdivision of the $t\times t$-wall is $d$-breakable.

The conjecture concerns induced-subgraph obstructions to small tree independence number and is presented as support for a broader program involving walls and domination-based separators. The source states that its theorem on $\mathcal{M}_t^*$-free graphs provides evidence, but does not specify a resolution.