Verstraëte's minimum-degree conjecture on cycle lengths in Hamiltonian graphs

Let $G$ be an $n$-vertex Hamiltonian graph, and let $\delta(G)$ denote its minimum degree. Verstraëte's conjecture. If

$$\delta(G)\geq 3,$$

then $G$ has $\Omega(n)$ different cycle lengths. This strengthens the Jacobson–Lehel question by replacing regularity with a minimum-degree condition. The paper proves the asymptotic lower bound $n^{1-o(1)}$, but the asserted linear bound remains open.