A homotopy rigidity theorem for ${\mathcal{Z}}_0$-stable ${\mathrm{C}}^{\ast }$-algebras

We show that two simple, separable, nuclear and ${\mathcal{Z}}_0$-stable ${\mathrm{C}}^{\ast }$-algebras are isomorphic if they are trace-preservingly homotopy equivalent. This result does not assume the UCT and can be viewed as a tracial stably projectionless analog of the homotopy rigidity theorem for Kirchberg algebras.