Conjecture on closures of beta-surfaces for singular self-dual Zollfrei metrics

Let $M$ be a four-manifold, let $C$ be a singular beta-surface, and let $g$ be a singular self-dual Zollfrei metric on $M$ with singular beta-surface $C$. Let $S$ be any beta-surface in $M-C$, and let $\overline{S}$ be its closure in $M$.

Closure conjecture. The set $\overline{S}-S$ is a finite subset of $C$, and $\overline{S}$ is a topological manifold.

This conjecture describes how beta-surfaces in the nonsingular region extend across the singular beta-surface. The authors motivate it by examples in which the closure is homeomorphic to $S^2$ with two additional points; the supplied text gives no resolution status.