Conjecture on compact CN2 manifolds with signed scalar curvature

A CN2 manifold is a Riemannian manifold whose curvature operator has nullity of codimension at most two. Let $M$ be a compact CN2 manifold with finite volume and scalar curvature that is everywhere nonnegative or everywhere nonpositive. Graph-manifold conjecture. Every such $M$ is a geometric graph manifold. The claim is stated as an expected consequence in the nonnegative or nonpositive scalar-curvature cases; the source provides no resolution here.