The regularity conjecture for simple nuclear C*-algebras

Let $A$ be a simple nuclear separable unital infinite-dimensional $C^*$-algebra. The properties of having finite nuclear dimension, being $\mathcal{Z}$-stable, and having strict comparison are defined for $A$ by the three conditions below. Regularity conjecture. The following conditions are equivalent:

$$\begin{array}{ll} (1) & A\text{ has finite nuclear dimension};\\ (2) & A\text{ is }\mathcal{Z}\text{-stable};\\ (3) & A\text{ has strict comparison}. \end{array}$$

The conjecture proposes equivalence among central regularity properties used in the classification of simple nuclear separable $C^*$-algebras. The implications $(1)\Longrightarrow(2)$ and $(2)\Longrightarrow(3)$ had been established by the third named author and Rørdam, respectively, while the reverse implications and hence the full equivalence were unresolved in the source.