The quadratic diameter conjecture for friends-and-strangers graphs on cycles

Let $Y$ be a graph on $n$ vertices, and let $\mathsf{FS}(\textsf{Cycle}_n,Y)$ denote its friends-and-strangers graph. The quadratic diameter conjecture asserts that the maximum diameter of a connected component of

$$\mathsf{FS}(\textsf{Cycle}_n,Y)$$

is $O(n^2)$. This would sharpen the paper's polynomial diameter bound for cycles; the conjecture is presented as an unresolved improvement, and would imply an $O(n^3)$ bound for the corresponding double-flip distances.