T-Winter conjecture on regularity properties of simple nuclear C*-algebras

Let $A$ be a simple unital nuclear separable $C^*$-algebra. The properties of finite nuclear dimension, $\mathcal{Z}$-stability, and strict comparison of positive elements are the three regularity properties considered below. T-Winter conjecture. The following are equivalent:

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

This conjecture extends the equivalence between $\mathcal{Z}$-stability and strict comparison known in the traceless case to simple unital nuclear separable $C^*$-algebras. The paper confirms the equivalence partially for simple AH algebras with strict comparison, but the full statement remains open in the source context.