The global spherical shell conjecture for class VII_0 surfaces

A class $\mathrm{VII}_0$ surface is a compact complex surface in class $\mathrm{VII}_0$, and $b_2$ denotes its second Betti number. A global spherical shell in a complex surface $X$ is an open subset $U$ diffeomorphic to a connected open subset of $S^3$ in $\mathbb{C}^2\setminus\{0\}$.

Global spherical shell conjecture. Every class $\mathrm{VII}_0$ surface $X$ with

$$b_2(X)>0$$

contains a global spherical shell; equivalently, it is a Kato surface. In the formulation given in the source, such surfaces are deformations of blown-up primary Hopf surfaces.

The conjecture is a partial converse to the fact that Kato surfaces are class $\mathrm{VII}$ surfaces with $b_1=1$. The condition $b_2>0$ and minimality are essential: Inoue surfaces have $b_2=0$ and do not contain global spherical shells, while blowing up an Inoue surface destroys minimality. The conjecture remains open in general.