Georgakopoulos–Papasoglu fat minor conjecture

Let $X$ and $G$ be graphs, let $K\in\mathbb{N}$, and let $f:\mathbb{N}\to\mathbb{N}^2$ be a function. A graph is $K$-fat-minor-free with respect to $X$ when it does not contain $X$ as a $K$-fat minor.

Fat minor conjecture. For every graph $X$ there exists a function $f:\mathbb{N}\to\mathbb{N}^2$ such that, for every graph $G$ and $K\in\mathbb{N}$, if $G$ does not contain $X$ as a $K$-fat minor, then $G$ is $f(K)$-quasi-isometric to a graph with no $X$ minor.

This conjecture has been disproved, with further counterexamples showing failure even for certain small graphs and under a weakened conclusion. It is therefore refuted rather than open.