Krupiński–Pillay's amenability conjecture for topological groups

Let $G(M)$ be a topological group. It is amenable when every $G(M)$-flow admits a $G(M)$-invariant Borel probability measure. Krupiński–Pillay's conjecture. If $G(M)$ is amenable, then

$$G^{00}_{\operatorname{top}}=G^{000}_{\operatorname{top}}.$$

This is Conjecture 0.2 of Krupiński and Pillay, connecting classical topological amenability with equality of model-theoretic topological components; the excerpt gives no resolution.