High-differentiability characteristic gluing conjecture

Let $k\in\mathbb{N}$, and let $\mathcal F$ be a compensating family of smooth metrics defined near $\mathcal N_2$. Let $x_1\in\Psi[\mathbf S_1,k]$ be a smooth, spacelike, vacuum, codimension-two data set. The data $x_1$ are sufficiently close in a suitable topology to the data induced by a member of $\mathcal F$.

Characteristic gluing conjecture. The data $x_1$ can be $C^k_u C^\infty_{(r,x^A)}$-glued to data induced on a deformation of $\mathbf S_2$ within a nearby member of $\mathcal F$.

This conjecture concerns characteristic gluing for vacuum data with arbitrarily prescribed finite transverse differentiability. The compensating family supplies the global charges needed to overcome the obstructions to direct gluing; the supplied text presents the assertion as the expected outcome of previous work, but gives no resolution status.