Purity and compact-support conjecture for self-similar states

Let $(A,L)$ satisfy the requirements of the strictly contractive dual iterated function system, and let $\phi_0\in\mathcal{S}_{\mathrm{c}}(A)$. Purity and compact-support conjecture. If $(A,L)$ is locally consistent, then every self-similar state $\phi_\omega$ is pure; if $(A,L)$ is locally semi-consistent, then every $\phi_\omega$ is compactly supported. These assertions concern the unresolved purity and support properties of self-similar states in the noncommutative setting; the surrounding discussion explains that both closedness of pure states and preservation of purity can fail or be subtle.