Strongly negative cases for Hamiltonicity in graphon random graphs

Let $W:\Omega^2\to[0,1]$ be a graphon, and let $\mathbb{G}(n,W)$ denote the associated inhomogeneous random graph. The four conditions in Proposition~ are: $W$ is not a connected graphon; $\lim_{\alpha\searrow 0}\mu(\mathsf{D}_\alpha(W))/\alpha=\infty$; $W$ has a narrow peninsula; or there is a partition $\Omega=S\sqcup T$ with $\mu(S)=\mu(T)=1/2$ such that $W$ is zero almost everywhere on $(S\times S)\cup(T\times T)$. Completeness conjecture. If $W$ satisfies none of these four conditions, then

$$\limsup_{n\to\infty}\mathbf{P}[\mathbb{G}(n,W)\text{ is Hamiltonian}]>0.$$

The conjecture asserts that these are exactly the strongly negative graphon obstructions, complementing the theorem that each listed condition makes Hamiltonicity asymptotically almost impossible. Its resolution would characterize when Hamiltonicity has probability bounded away from zero rather than tending to zero.