Weak pure infiniteness and O-infinity absorption

We prove that every complex C${}^{\ast }$-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam's comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs ${\mathcal{O}}_{\infty }$, without assumptions of unitality or simplicity.