  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.
