We prove that, if a topological group G has an open subgroup of infinite index, then every net of tight Borel probability measures on G UEB-converging to invariance dissipates in G in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric d on the infinite symmetric group Sym(ℕ), compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups (Sym(n), d ↾_Sym(n), μ_Sym(n))_n∈ℕ equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov’s observable distance.