  We show that every nuclear $\mathcal O_\infty$-stable *-homomorphism with a
separable exact domain has nuclear dimension at most 1. In particular
separable, nuclear, $\mathcal O_\infty$-stable C*-algebras have nuclear
dimension 1. We also characterise when $\mathcal O_\infty$-stable C*-algebras
have finite decomposition rank in terms of quasidiagonality and primitive-ideal
structure, and determine when full $\mathcal O_2$-stable *-homomorphisms have
nuclear dimension 0.
