Equivariant concentration in topological groups

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and $({\mu }_n{)}_{n\in \mathbb{N}}$ is a sequence of compactly supported Borel probability measures on $G$ converging to invariance with respect to the mass transportation distance over $d$ and such that $\left($\mathrm{s}\mathrm{p}\mathrm{t}$ , \mu_{n}, d!!\upharpoonright_{$\mathrm{s}\mathrm{p}\mathrm{t}$ , \mu_{n}}, \mu_{n}!!\upharpoonright_{$\mathrm{s}\mathrm{p}\mathrm{t}$ , \mu_{n}}\right){n \in $\mathbb{N}$}$concentratestoafullysupported,compact$mm$-space$\left(X,d{X},\mu_{X}\right)$,then$X$ishomeomorphictoa$G$-invariant subspace of the Samuel compactification of $G$. In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov’s observable diameters along any net of Borel probability measures UEB-converging to invariance over the group.