The bounded-defect cycle conjecture for Hamiltonian graphs

Let $G$ be an $n$-vertex graph, and write $\delta(G)$ for its minimum degree. A Hamiltonian cycle is a cycle spanning all vertices of $G$. The bounded-defect cycle conjecture. If $\delta(G)\geq 3$ and $G$ contains a Hamiltonian cycle, then $G$ contains another cycle of length at least

$$n-K,$$

where $K>0$ is an absolute constant. The conjecture asks whether the error term in the paper's asymptotic result can be bounded independently of $n$; the source notes that even $K=2$ may be possible, while related constructions may rule out such small values.