The staircase representation conjecture for Cheeger graphs

Let $G_2(1)$ be the indicated two-vertex graph, let $\sqcup$ denote disjoint union, and let a staircase graph mean a graph of the staircase type defined in the paper. Staircase representation conjecture. Every Cheeger graph other than

$$G_2(1)\sqcup G_2(1)$$

can be represented as a staircase graph. This daring conjecture would imply both the triangle-freeness and bipartiteness conjectures for Cheeger graphs. It is presented as an open problem.