https://arxiv.org/pdf/1503.08825
The first comment in figure 1. seems to be wrong
' Even in the presence of a black hole, nice slices can be parameterized by coordinates on $ [0, 1) \times S^{d - 1} $ '
It is clear that the nice slices unambiguously specify the CFT operators. But it will not give the generalized harmonic gauges.
If you use the full tensor $ g_{ij} $, we run into difficulties with gauge invariance. This can be avoided by taking a scalar field $g_{00}$
This is gauge invariant because for a scalar field the $ \hbar $ in the Klein-Gordon equation can be removed.