The logarithmic-degree conjecture for cycles with linearly many chords

Let $G$ be a graph on $n$ vertices, and let $C$ be a positive constant. A cycle with linearly many chords is a cycle on $[200~\ell$ vertices having at least $[201~\ell/2$ chords.

Cycle-with-chords conjecture. If $G$ has average degree at least $C\log\log(n)$, then $G$ contains a cycle $C$ on $[202~\ell$ vertices with at least $[203~\ell/2$ chords, for some $[204~\ell\geqslant 4$.

The surrounding discussion presents this as a conjectural strengthening of the paper's results on cycles with almost linearly many chords; it also suggests that the regular-graph case may be a natural first step. The source gives no evidence that the conjecture has been resolved.