Noncompact Bonnet–Myers conjecture for Q-curvature and scalar curvature

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. Noncompact Bonnet–Myers conjecture. If

$$Q_g\geq 6 \quad\text{and}\quad R_g\geq 0,$$

then $M^4$ is compact. This extends the corresponding proposition from compact four-manifolds. The claim is currently unproved in general, although it is established when the Ricci curvature is bounded from below; in that case one obtains $R_g\geq 12$ and then compactness by the Bonnet–Myers-type theorem.