The strong conjecture on anti-GMS sets in Polish groups

Let $\mathbb{G}$ be a Polish group. A nonempty subset $C\subseteq\mathbb{G}$ is anti-GMS if it is nowhere dense and, for every sequence $\{U_n:n\in\omega\}$ of open neighborhoods of the identity, there is a sequence $\{g_n:n\in\omega\}$ in $\mathbb{G}$ such that, for every $g\in\mathbb{G}$, the set $g\cdot\bigcup_{n\in\omega}g_n\cdot U_n$ is dense in $C$. Strong conjecture. Exactly one of the following holds for any Polish group $\mathbb{G}$: either $\mathbb{G}$ is locally compact, or it contains an anti-GMS set. Anti-GMS sets are the paper's tool for disproving the Galvin–Mycielski–Solovay theorem in non-locally compact groups; the dichotomy itself remains open.