Topological universality conjecture for locally compact Polish groups

Let $\mathbb{G}$ be a locally compact Polish group, and call a subset of $\mathbb{G}$ topologically universal if every dense $G_\delta$ subset of $\mathbb{G}$ contains its image under an affine transformation. Locally compact Polish-group conjecture. There is no uncountable topologically universal set in any locally compact Polish group. The paper states that the topological variant is resolved in locally compact Polish groups; the supplied passage gives the conjecture's formulation but not the proof.