Approaching the UCT problem via crossed products of the Razak-Jacelon algebra

We show that the UCT problem for separable, nuclear $\mathrm C^*$-algebras relies only on whether the UCT holds for crossed products of certain finite cyclic group actions on the Razak-Jacelon algebra. This observation is analogous to and in fact recovers a characterization of the UCT problem in terms of finite group actions on the Cuntz algebra $\mathcal O_2$ established in previous work by the authors. Although based on a similar approach, the new conceptual ingredients in the finite context are the recent advances in the classification of stably projectionless $\mathrm C^*$-algebras, as well as a known characterization of the UCT problem in terms of certain tracially AF $\mathrm C^*$-algebras due to Dadarlat.