The paper disproves the conjecture by Georgakopoulos and Papasoglu that a length space (or graph) without a $K$-fat $H$ minor is quasi-isometric to a graph without an $H$ minor. Specifically, it provides a counterexample where the graph $G_q$ does not contain $H$ as a 3-fat minor and is not $q$-quasi-isometric to any graph with no (2-fat) $H$ minor. The authors also show that graphs without a $K$-fat $H$ minor are quasi-isometric to graphs without a 3-fat $H$ minor, up to a scaling factor. The proof involves a construction based on a recent counterexample to a coarse version of Menger's theorem by Nguyen, Scott, and Seymour. The paper concludes with remarks on the construction and weaker conjectures that might hold in the coarse setting.The paper disproves the conjecture by Georgakopoulos and Papasoglu that a length space (or graph) without a $K$-fat $H$ minor is quasi-isometric to a graph without an $H$ minor. Specifically, it provides a counterexample where the graph $G_q$ does not contain $H$ as a 3-fat minor and is not $q$-quasi-isometric to any graph with no (2-fat) $H$ minor. The authors also show that graphs without a $K$-fat $H$ minor are quasi-isometric to graphs without a 3-fat $H$ minor, up to a scaling factor. The proof involves a construction based on a recent counterexample to a coarse version of Menger's theorem by Nguyen, Scott, and Seymour. The paper concludes with remarks on the construction and weaker conjectures that might hold in the coarse setting.