Asymptotics of maximum capture time at larger speeds

Determine the asymptotic behavior of capt^*_s(n) for larger speeds s, including whether there exists a function f(s) such that capt^*_s(n)=n−f(s) in general.

Background

The paper constructs speed-(2,2) examples with capture time at least n−7 and reports computational evidence suggesting that the analogous maximum capture time for speed 3 is n−O(1). It leaves unresolved both whether this behavior holds for speed 3 and what the general dependence on n and s should be for larger speeds.

References

From computer search of small graphs, we believe that $capt_{3}*(n)=n-O(1)$; it would be interesting to verify (or refute) this. More generally, what are the asymptotics of $capt_{s}*(n)$ for larger $s$? Is it perhaps the case that, in general, $capt_{s}*(n) = n - f(s)$ for some function $f$?

— Accelerated Cops and Robbers  (2506.20753 - Kinnersley et al., 25 Jun 2025) in Section Future Work, seventh bullet