Fat minors cannot be thinned (by quasi-isometries)

We disprove the conjecture of Georgakopoulos and Papasoglu that a length space (or graph) with no K-fat H minor is quasi-isometric to a graph with no H minor. Our counterexample is furthermore not quasi-isometric to a graph with no 2-fat H minor or a length space with no H minor. On the other hand, we show that the following weakening holds: any graph with no K-fat H minor is quasi-isometric to a graph with no $3$-fat H minor.