Nakamura's GSS conjecture for class VII$_0^+$ surfaces

A compact complex surface in class VII$_0^+$ is a surface with first Betti number one and positive second Betti number. A Kato surface is a compact complex surface obtained by Kato's construction and possessing a global spherical shell (GSS). Nakamura's GSS conjecture. Every surface in class VII$_0^+$ is a Kato surface. This conjecture would classify all compact complex surfaces in class VII$_0^+$ by showing that the Kato surfaces are the only ones; the statement remains open.