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

Let $A$ be a separable, simple, nuclear, unital, infinite-dimensional $C^*$-algebra. Nuclear dimension is a noncommutative analogue of covering dimension;

$$denotes the Jiang--Su algebra, and strict comparison is the standard comparison property for positive elements. **Toms--Winter Conjecture.** The following conditions are equivalent:$$

A \text{ has finite nuclear dimension},

$$$$

A \text{ is } \text{-stable},

$$$$

A \text{ has strict comparison}.

$$The equivalence of finite nuclear dimension and$$

-stability is known, and both imply strict comparison. Thus the remaining implication is the unresolved part of the conjecture in this formulation.