Petrunin–Tuschmann conjecture on simply connected ends of conical asymptotically flat 4-manifolds

Let $X$ be a conical $\mathcal{AF}$ 4-manifold with a simply connected end, and let its asymptotic cone be the metric cone at infinity. Petrunin–Tuschmann conjecture. There is no conical $\mathcal{AF}$ 4-manifold with a simply-connected end whose asymptotic cone is the half plane.

This conjecture concerns the only remaining asymptotic cone allowed by the Petrunin–Tuschmann structural result in dimension four that is not already known to be realized by a simply connected end. The half plane is realized by examples with non-simply-connected ends, but the simply connected case remains open.