Improve the cyclic clique-minor bound

Determine whether the function 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

The paper defines f(k) as the minimum degree threshold guaranteeing a cyclic K_k-minor. It establishes f(4)=3, 6\leq f(5)\leq 8, and the general upper bound f(k)=O(k2). The authors ask whether the stronger order O(k\sqrt{\log k}) is possible, which would match the known asymptotic bound for ordinary clique minors.

References

We can prove that $f(4)=3$, and $6 \le f(5)\le 8$ and more generally $f(k)=O(k2)$. 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 Section 1, subsection “Dense Cyclic minors”