Hamiltonicity conjecture for weakly pseudorandom graphs

Let a weakly $(n,\beta)$-graph be a graph in the sense defined in the source, and let $G$ be Hamiltonian when it contains a Hamilton cycle. Hamiltonicity conjecture for weakly $(n,\beta)$-graphs. There exists a constant $\beta>0$ such that every weakly $(n,\beta)$-graph with $n\ge 3$ (or sufficiently large $n$) is Hamiltonian. The source says this conjecture is stronger than the stated Krivelevich–Sudakov and Gu conjectures; its resolution is not given.