Hilton's cycle multiplicity conjecture for dense hamiltonian graphs

Let $G$) be a hamiltonian graph of order $n$, and let $e(G)$ denote its number of edges and $c_\ell(G)$ the number of cycles of length $\ell$ in $G$. Assume

$$e(G) > \left\lfloor \frac{n^2}{4} \right\rfloor + 1.$$

Hilton's conjecture. For every integer $\ell$ with $3\le \ell\le n$,

$$c_\ell(G) \ge n-\ell+2.$$

The conjecture strengthens Sheehan's theorem, which guarantees at least two cycles of every length under the same density condition. It is known at the endpoint $\ell=n$, and the triangle case follows from a result of Erdős; the full conjecture has remained open since 1977, although the paper proves it for sufficiently large order and leaves only finite exceptional ranges for lengths $6\le \ell\le 10$.