The triangle-freeness conjecture for Cheeger graphs

Let $G$ be a Cheeger graph, meaning a graph attaining the first Cheeger constant of the relevant simplex. Triangle-freeness conjecture. All Cheeger graphs are triangle-free. This is motivated by the authors' examples, in which an optimal graph's Cheeger constant receives no contribution from a triangle whose three edges all belong to the graph. The conjecture is presented as an open problem.