Burns–Knieper flatness conjecture for bounded-distance geodesic foliations

Let $M$ be a simply connected surface with a complete Riemannian metric with no conjugate points. Suppose that $\mathcal F$ is a foliation on $M$ whose leaves are all geodesic lines, and any two of which are at finite Hausdorff distance. Burns–Knieper's conjecture. Then $M$ is flat.

The conjecture asks whether a simply connected Riemannian plane without conjugate points must be flat when it admits a geodesic foliation whose leaves remain at finite Hausdorff distance from one another. It is proved in the source under the additional hypotheses that the plane admits total curvature or has visibility at some point; the general case remains open.