Linear edge threshold for cycles with at least |C| chords

Determine whether there exists a constant c > 0 such that every n-vertex graph with at least c · n edges contains a cycle C with at least |C| chords.

Background

Draganić, Methuku, Munhá-Correia, and Sudakov proved that c * n (log n)^8 edges suffice, which is tight up to the polylogarithmic factor. Removing the logarithmic factors would settle whether a linear edge threshold guarantees the existence of cycles with many chords.