Gallagher–Lai–Weber topological Erdős similarity conjecture

Let $X\subseteq\mathbb{R}$ be topologically universal if every dense $G_\delta$ subset of $\mathbb{R}$ contains an affine copy of $X$. Gallagher–Lai–Weber's conjecture. No uncountable subset of the real line is topologically universal. Countable sets are topologically universal by the Baire category theorem, and perfect sets are not; the conjecture is resolved in this paper for $\mathbb{R}$, although the source does not state the resolution in the supplied span.