Amenability of the component group of the unitary group

Let $A$ be a nuclear $C^*$-algebra. Its unitary group is denoted by ${\rm U}(A)$, and ${\rm U}_0(A)$ is the connected component of the identity. The quotient ${\rm U}(A)/{\rm U}_0(A)$ is viewed as a discrete group.

Component-group amenability conjecture. The discrete group

$${\rm U}(A)/{\rm U}_0(A)$$

is amenable.

This would follow from the preceding equivalence conjecture and is known in many natural classification-theoretic examples where the quotient is isomorphic to the abelian group $K_1(A)$. The general assertion remains open, and the source notes that the quotient can nevertheless be non-abelian in natural examples.