Bounded-diameter Yamabe sequence conjecture for Kodaira surfaces

Let $M$ be the underlying 4-manifold of a complex algebraic surface of Kodaira dimension $\geq 0$. A sequence of unit-volume constant-scalar-curvature metrics on $M$ has uniformly bounded diameter if its diameters are uniformly bounded. Bounded-diameter Yamabe sequence conjecture. If there is such a sequence with scalar curvature $s\nearrow 0$, then $\operatorname{Kod}(M)=0$. This would distinguish Kodaira dimension $0$ from $1$ for non-minimal surfaces; the source gives no resolution.