Krivelevich–Sudakov Hamiltonicity conjecture for pseudorandom graphs

Let $G$ be an $(n,d,\lambda)$-graph, meaning an $n$-vertex $d$-regular graph whose non-trivial eigenvalues have absolute value at most $\lambda$. A Hamilton cycle is a cycle containing every vertex of $G$. Krivelevich and Sudakov's conjecture. There exists a positive constant $C$ such that, if

$$\frac{d}{\lambda}\ge C,$$

then $G$ contains a Hamilton cycle.

This conjecture asks for a spectral-gap condition guaranteeing Hamiltonicity in pseudorandom graphs. The paper presents it as a conjecture posed nearly 20 years ago; no resolution is supplied in the provided context.