Yamabe invariant conjecture for compact complex surfaces of Kodaira dimension 0 or 1

Let $M$ be the underlying 4-manifold of a compact complex manifold of Kodaira dimension $0$ or $1$, and let $b_1(M)$ denote its first Betti number. Yamabe invariant conjecture for Kodaira dimension 0 or 1. Even if $b_1(M)$ is odd, one still has

$$Y(M)=0.$$

The conjecture extends the known conclusion for even first Betti number to the odd case. The source notes that Kodaira surfaces have non-trivial Seiberg–Witten invariants, while the situation for Kodaira dimension $1$ is less clear; no resolution is supplied.