Gu's Laplacian spectral Hamiltonicity conjecture

Let $G$ be the graph under consideration, with Laplacian eigenvalues $\mu_2$ and $\mu_n$, and say that $G$ is Hamiltonian when it contains a Hamilton cycle. Gu's conjecture. There exists a constant $C$ with $0<C<1$ such that

$$\frac{\mu_2}{\mu_n}\ge C$$

and $n\ge 3$ (or, alternatively, for sufficiently large $n$) imply that $G$ is Hamiltonian. The source presents this as a stronger Laplacian-eigenvalue analogue of the Krivelevich–Sudakov conjecture and does not report a resolution.