I will talk about my recent work with Bhishan Jacelon and Luis Santiago on the stably projectionless C*-algebra W, previously studied by Bhishan Jacelon. We show that W is tensorially self-absorbing. We also show that tensorial absorption of W entails strong regularity properties for a given C*-algebra. Our results lend evidence to the conjecture that amenable W-absorbing C*-algebras are classified by their cones of traces.