The Galvin–Mycielski–Solovay characterization of locally compact Polish groups

Let $\mathbb{G}$ be a Polish group. A set $S\subseteq\mathbb{G}$ is of strong measure zero, and a set $M\subseteq\mathbb{G}$ is meager. Galvin–Mycielski–Solovay characterization conjecture. The Galvin–Mycielski–Solovay theorem holds in $\mathbb{G}$ if and only if $\mathbb{G}$ is locally compact. The source presents this as a stronger $\mathsf{ZFC}$ conjecture related to anti-GMS sets; the locally compact direction is a theorem, while the converse is not established in general.