Existence of higher-regularity subextremal gluing solutions

Establish whether the existence theorem for subextremal $C^0$ gluing solutions also holds for $C^k$ gluing solutions, for higher regularity classes $C^k$ as contemplated in the paper.

Background

The paper proves that for every subextremal Reissner–Nordström horizon with $|Q_+|<r_+$ and every nonzero scalar charge $\mathfrak{e}$, there exists a $C^0$ gluing solution connecting a Schwarzschild exterior cone to the subextremal horizon. A remark explains that, with modifications such as smoothing the scalar-field transitions and enforcing an additional integral constraint, these constructions can be upgraded to $C^1$ solutions. The paper leaves unresolved whether the corresponding existence result holds for arbitrary higher regularity classes $C^k$.