The isometry group of global $AdS_{d+1}$ is well-known to be $SO(d,2)$. I have a suspicion that when the spacetime is asymptotically AdS, with dynamical gravity in the bulk, the symmetry group gets enhanced to the Weyl group. This would be analogous to how the Poincare group gets enhanced to the BMS symmetry group of asymptotically flat spacetime. I'd like to know if my suspicion is correct.
My rough reasoning is as follows. Let us write the asymptotically AdS metric in Fefferman-Graham form:
$$ \tag{1} ds^2 = \frac{l^2}{r^2}\left(dr^2 + (g^{(0)}_{ij}+ r g^{(1)}_{ij} + ... + r^d g^{(d)}_{ij}+...)dx^idx^j \right),$$
where the Einstein equations fix all other coefficients in terms of $g_{ij}^{(0)}$ and $g_{ij}^{(d)}$.
In the asymptotically flat case, the BMS symmetry group consists of transformations that leave the asymptotic form of the metric invariant. Likewise, in the AdS case there is a subgroup of diffeomorphisms (the so-called PBH transformations) which leave the metric in FG form but with new coefficients. In particular, they act on $g_{ij}^{(0)}$ as Weyl-recalings. Therefore, I'd like to conclude that gravity in asymptotically AdS has Weyl group symmetry.
My issue is that I don't truly know what "symmetry group" means in this context. Since I study QFT, I want to consider symmetries of the gravitational path integral viewed as a function of the boundary data $g_{ij}^{(0)}$ and $g_{ij}^{(d)}$. More concretely, to say that the theory has Weyl symmetry, I want the path integral to be invariant under Weyl-rescaling both the "background" $g_{ij}^{(0)}$ and the "dynamical field" $g_{ij}^{(d)}$. The problem is, I don't see how this type of symmetry is related to the notion of "leaving the asymptotic form of the metric invariant".
So, what is the appropriate notion of "symmetry group" in an asymptotically AdS theory of gravity, and why? And is it correct that this symmetry group is enhanced from $SO(d,2)$ to Weyl, just as Poincare is enhanced to BMS in the flat case?