Conjecture that the generic fiber is deformation equivalent to a Barlow surface

Let $X^{\varepsilon}$ be the generic fiber of the degeneration constructed in the paper, and let a deformation class mean an equivalence class of surfaces connected by a deformation. The Barlow-surface conjecture. The generic fiber $X^{\varepsilon}$ is in a deformation class of Barlow's surface. This conjecture identifies the constructed generic fiber with the deformation type of a Barlow surface; the supplied text does not state whether it has been resolved.