Equivalence of pure infiniteness properties for separable nuclear C*-algebras

Determine whether weak pure infiniteness, pure infiniteness, and strong pure infiniteness are equivalent for every separable nuclear C*-algebra.

Background

The paper introduces two notions for essential extensions into corona algebras: pure infiniteness, defined through Cuntz subequivalence of two copies of each nonzero positive image, and strong $O_{\infty}$-stability, defined through asymptotically central pairs of isometries. The authors explain that these notions correspond roughly to pure infiniteness and $O_{\infty}$-stability for C*-algebras.

They note that, although the three standard pure infiniteness notions coincide in the simple case, their equivalence for arbitrary separable nuclear C*-algebras is unresolved. This question concerns the relationship among weak pure infiniteness, pure infiniteness, and strong pure infiniteness in the nonsimple setting.