Kuratowski-type coarse planarisation conjecture

A graph has a $K$-fat $K_5$ and $K_{3,3}$ minor if it contains both $K_5$ and $K_{3,3}$ as fat minors with parameter $K$. Let

$$f \colon \mathbb{N} \to \mathbb{N} \times \mathbb{N}.$$

Kuratowski-type coarse planarisation conjecture. There exists a function $f$ such that every graph with no $K$-fat $K_5$ and $K_{3,3}$ minor is $f(K)$-quasi-isometric to a planar graph.

This is described as a coarse variant of Kuratowski's theorem and as a special case of the broader Georgakopoulos–Papasoglu conjecture. The paper states that this special case remains open.