Girao, Kittipassorn, and Narayanan's long-cycle conjecture

Let $G$ be a simple graph with $n$ vertices and minimum degree $\delta(G)\geq 3$, and suppose that $G$ contains a Hamiltonian cycle. Girao, Kittipassorn, and Narayanan's conjecture. There exists a constant $c>0$ such that $G$ contains another cycle of length at least $n-c$. Girao, Kittipassorn, and Narayanan proved a polynomial-error version with loss $cn^{4/5}$; the paper improves this to loss $cn^{2/3}$, but the constant-error assertion remains open.