Sobriety of the Scott space of the rational line

Determine whether the Scott space of the lattice of open subsets of the rational line \(\mathbb{Q}\) is sober, equivalently, whether the Hoare power space \(H(\mathbb{Q})\) is co-sober.

Background

The paper establishes a general equivalence for every $T_0$-space $X$: the Hoare power space $H(X)$ is co-sober if and only if the Scott space of the lattice $\mathcal{O}(X)$ of open subsets of $X$ is sober. The authors then note that He and Zhao posed the specific question of whether this Scott space is sober when $X=\mathbb{Q}$, the rational line. The construction developed in the paper produces a different countable metrizable counterexample and therefore does not resolve the rational-line case.