Cocomparability of the next graph power

Determine whether the $(k+1)$-st power of a graph is necessarily cocomparability whenever its $k$-th power is AT-free.

Background

The paper establishes that if a graph has bounded path-length, then a bounded power of the graph is AT-free and, separately, that another bounded power is cocomparability. The authors identify an unresolved implication between these two graph classes: although AT-freeness of a power is known, it is not known whether increasing the power by one always yields a cocomparability graph. The subsequent lemma provides a different cocomparability bound, namely that the 2λ2\lambda-th power is cocomparability when the graph has path-length at most λ\lambda, but it does not settle the stated one-step question.

References

We do not know if $G{k+1}$ is a cocomparability graph for a graph $G$ whose $k{th}$ power $Gk$ is an AT-free graph.

— Graph parameters that are coarsely equivalent to path-length  (2503.05661 - Dragan et al., 7 Mar 2025) in Section 3.1, immediately before Lemma 3.2