The cubic-scale diameter conjecture for 4-configuration graphs

Let $D(n,k)$ denote the maximum diameter of a connected component of the $k$-configuration graph over all graphs on $n$ vertices. The $k$-configuration graph $\mathcal{R}_k(G)$ has independent $k$-sets of $G$ as vertices, with adjacency when one token is moved along an edge while preserving independence.

Cubic-scale diameter conjecture.

$$D(n,4)=n^{3-o(1)}.$$

The paper notes that the lower and upper bounds are not known to almost match for larger $k$; in particular, it is open whether the 4-configuration graph can have super-quadratic diameter, while the known upper bound is $o(n^3)$.