Negative Yamabe invariant conjecture for compact complex surfaces

Let $(M,J)$ be a compact complex surface, where $M$ is its underlying 4-manifold and $Y(M)$ is its Yamabe invariant. Negative Yamabe invariant conjecture. If

$$Y(M)<0,$$

then $M$ is of general type. The source explains that the theorem immediately preceding this conjecture allows surfaces of type VII as an additional possibility, but that all known type VII surfaces have $Y\geq 0$; the conjecture remains open for the potentially unknown type VII surfaces.