The equivalence of nuclearity, symmetric amenability, strong amenability and unitary-group amenability

Let $A$ be a unital, separable $C^*$-algebra. The QTS property means that every non-trivial quotient of $A$ admits a non-trivial trace. The unitary group of $A$ is denoted by ${\rm U}(A)$, and ${\rm U}_0(A)$ denotes its connected component of the identity. A topological group is amenable if its bounded right-uniformly continuous functions admit a left invariant mean, and skew-amenable if they admit a right invariant mean.

Amenability equivalence conjecture. The following conditions are equivalent:

1. $A$ is nuclear and has the QTS property.
2. $A$ is symmetrically amenable.
3. $A$ is strongly amenable.
4. ${\rm U}(A)$ is amenable in the norm topology.
5. ${\rm U}(A)$ is skew-amenable in the weak topology.

Nuclearity is already known to characterize weak-topology amenability of the unitary group, while symmetric amenability is known to be equivalent to nuclearity together with the QTS property. The conjecture proposes that these properties also coincide with strong amenability and the stated norm- and weak-topology amenability properties of the unitary group.