Generalized Toms--Winter Conjecture for nuclear C*-algebras

Let $A$ be a separable, unital, nuclear $C^*$-algebra with no elementary subquotients, where a subquotient means a quotient of an ideal and an elementary $C^*$-algebra is one isomorphic to $\mathcal{K}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$. Let purity mean the regularity property used for nuclear $C^*$-algebras.

Generalized Toms--Winter Conjecture. The following conditions are equivalent:

$$A \text{ has finite nuclear dimension},$$ $$A \text{ is } \text{-stable},$$ $$A \text{ is pure}.$$

The simple case is motivated by the Toms--Winter conjecture. The supplied context gives related equivalences and results for graph algebras, but does not state a resolution of this generalized conjecture.