On equivalence relations induced by locally compact TSI Polish groups admitting open identity component

For a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^{\omega }\mathrm{/}c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. We prove a Rigid theorem on locally compact TSI Polish groups admitting open identity component, as follows: Let $G$ be a locally compact TSI Polish group such that $G_0$ is open in $G$, and let $H$ be a nontrivial pro-Lie TSI Polish group. Then $E(G){\le }_{B}E(H)$ iff there exists a continuous homomorphism $\phi :G_0\to H_0$ satisfying the following conditions: (i) $\ker (\phi )$ is non-archimedean; (ii) $\phi {\mathrm{I}\mathrm{n}\mathrm{n}}_G(G_0)\subseteq \overline{{\mathrm{I}\mathrm{n}\mathrm{n}}_H(H_0)\phi }$ under pointwise convergence topology. An application of the Rigid theorem yields a negative answer to Question 7.5 of [2].