Asymptotic bound for cyclic clique minors

Determine whether the minimum degree threshold f(k), defined as the smallest integer such that every graph of minimum degree at least f(k) contains K_k as a cyclic minor, satisfies f(k)=O(k\sqrt{\log k}).

Background

For a positive integer k, f(k) is defined as the smallest minimum-degree threshold guaranteeing that every graph contains the complete graph K_k as a cyclic minor. The paper establishes f(4)=3, 6≤f(5)≤8, and the general upper bound f(k)=O(k2). The authors ask whether the quadratic bound can be improved to O(k√(log k)), which would match the known asymptotic bound for ordinary clique minors.

References

Could it be that $f(k)=O(k\sqrt{\log k})$, matching the bound for normal minors from?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Section 1, subsection “Dense cyclic minors”

We propose the following open question. Could it be that $f(k)=O(k\sqrt{\log k})$, matching the bound for normal minors from?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Introduction, subsection “Dense cyclic minors”