The quadratic double-flip distance conjecture for acyclic orientations

Let $G$ be an $n$-vertex graph, and let $\alpha,\alpha”\in\operatorname{Acyc}(G)$ be double-flip equivalent, meaning that they can be connected by a sequence of allowed double-flips. The quadratic double-flip distance conjecture asserts that it is possible to go from $\alpha$ to $\alpha”$ in $O(n^2)$ double-flips. This is described as an unresolved strengthening of the paper's existing distance bound; together with the preceding cycle-diameter conjecture, it concerns efficient connectivity within double-flip equivalence classes.