Let $G$ be a locally compact (second countable) group and let $$ G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}. $$ This is the kernel of the Bohr compactification $G \to bG$, and in particular a closed normal subgroup. Suppose that $G/G_0$ is compact (so that $bG = G/G_0$).

Q: Is it true that $G_0$ is minimally almost periodic, that is, admits no non-trivial finite-dimensional unitary representations?

If $G/G_0$ is finite, then the answer is positive, because in this case a finite-dimensional representation of $G_0$ induces a finite-dimensional representation of $G$.