Fau's graph-algebra nowhere-scatteredness conjecture

Let $E$ be a row-finite graph, and let $C^*(E)$ denote its graph $C^*$-algebra. The algebra $C^*(E)$ is nowhere scattered when it has no nonzero scattered quotient. The graph $E$ is said to have Condition (K) and distinct detours as in the source.

Fau's graph-algebra conjecture. $C^*(E)$ is nowhere scattered if and only if $C^*(E)$ is $\mathcal{Z}$-stable, if and only if $E$ has Condition (K) and distinct detours.

This is a characterization of nowhere scatteredness for graph $C^*$-algebras in terms of regularity of the algebra and combinatorial properties of the graph. The supplied text gives a related theorem for finite graphs, but does not establish the stated row-finite formulation.