Flatness conjecture for Type IIb scalar-flat Kähler 4-manifolds

Let $M^4$ be a complete scalar-flat Kähler 4-manifold with a continuous symmetry, of Type IIb, and let $V$ denote its Killing field. Assume that $|V|$ has a global lower bound

$$|V|>\epsilon>0.$$

Flatness conjecture. Then $M^4$ is flat.

This conjecture concerns the case in which the level sets of the symmetry function are non-compact. It would classify Type IIb metrics whose Killing field is bounded away from zero; the source compares it with the Ricci-flat case and with results for manifolds having two symmetries.