We narrow the gap between the family of graphs that do and the family of graphs that do not satisfy the fat minor conjecture by obtaining much simpler counterexamples than were previously known, including $K_t, t \geq 6$ and $K_{s,t}, s,t \geq 4$ and $K_{2,2,2}$. This is achieved by establishing a `coarse self-similarity' property of the graphs used by Nguyen, Scott and Seymour to disprove the `coarse Menger conjecture'. This property may be of independent interest.