The bounded-cycle-space fat minor conjecture

Let $X$ be a graph, and let $G$ be a graph whose cycle space is generated by cycles of length at most $\kappa$. A graph is $K$-fat-minor-free with respect to $X$ when it does not contain $X$ as a $K$-fat minor.

Bounded-cycle-space fat minor conjecture. For every graph $X$ there exists a function $f:\mathbb{N}^2\to\mathbb{N}^2$ such that, for every graph $G$ whose cycle space is generated by cycles of length at most $\kappa$, if $G$ does not contain $X$ as a $K$-fat minor for some $K\in\mathbb{N}$, then $G$ is $f(\kappa,K)$-quasi-isometric to a graph with no $X$ minor.

This is presented as a still-open special case of the refuted fat minor conjecture. The bounded-cycle-space hypothesis is significant because the known counterexamples do not have cycle spaces generated by cycles of bounded length.