Unbounded chord growth for minimum-degree-three graphs

Determine whether there exists a function f tending to +∞ such that every graph of minimum degree 3 contains a cycle of length ℓ with at least f(ℓ) chords.

Background

The authors state that the preceding linear lower-bound problem is far from being resolved and formulate a weaker relaxation. This relaxation asks whether the chord count of a cycle in every minimum-degree-three graph can be forced to grow without bound as a function of the cycle length, rather than linearly.

References

Is there a function $f$ that tends to $+\infty$ such that every graph with minimum degree 3 contains a cycle of length $\ell$ with at least $f(\ell)$ chords?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Section 5, “Concluding remarks and open problems”

We are very far from answering the previous problem, as even the following relaxation seems unclear: Is there a function $f$ that tends to $+\infty$ such that every graph with minimum degree 3 contains a cycle of length $\ell$ with at least $f(\ell)$ chords?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Section 5, “Concluding remarks and open problems”