The Toms–Winter conjecture

Let $A$ be a unital, separable, simple, nuclear, non-elementary $C^*$-algebra. Say that $A$ has finite nuclear dimension when its nuclear dimension is finite, that $A$ is $\mathcal{Z}$-stable when $A\otimes\mathcal{Z}\cong A$, and that $A$ has strict comparison when positive elements are compared by lower semicontinuous $2$-quasitraces as described in the source.

The Toms–Winter conjecture. The following conditions are equivalent:

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

The conjecture arose in the Elliott classification program and asserts that three central regularity properties of simple nuclear $C^*$-algebras coincide. The supplied text does not provide evidence resolving the conjecture, so its status is recorded as open.