Lacunary sets for actions of tsi groups

Under a mild definability assumption, we characterize the family of Borel actions $\Gamma \curvearrowright X$ of tsi Polish groups on Polish spaces that can be decomposed into countably-many actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of $1_\Gamma$. In the special case that $\Gamma$ is non-archimedean, it follows that there is such a decomposition if and only if there is no continuous embedding of $\mathbb{E}_0^{\mathbb{N}}$ into $E_\Gamma^X$.