Why do we not try to describe $4$d quantum gravity with a $3$d CFT? In AdS/CFT correspondence, one mostly studies the case with AdS$_5 \times$ S$_5$ on the string side and $4$d $\mathcal{N}=4$ Super Yang-Mills on the gauge field theory side. Real-world observations show that gravity occurs in a $4$-dimensional Universe, possibly with compact dimensions. Then would it not be more interesting to look at correspondences between AdS$_4$ and $3$d CFT? In that way, one would hope to be able to describe $4$d quantum gravity with a $3$d Yang-Mills theory or something. Maybe the reason is that this implies giving up on the Standard Model, but when I think about it I do not see an obvious reason why this should be necessary at first (maybe in latter stages though).
Note that I am putting aside for now the fact that AdS space does not seem to reproduce our Universe. Also, I am aware that there actually exists such dualities in the literature, but they seem more academic-oriented, in the sense that their aim is not to describe reality (maybe I am wrong?). And of course I can imagine the $\mathcal{N}=4$ being useful for practical purposes if one tries to describe e.g. QCD with string theory.
 A: It is crucial to understand that in the $AdS_{5} \times S^{5}$ realization of holography spacetime is percieved as ten dimensional by any bulk observer because the radius of $S^{5}$ is macroscopic (much larger that the string lenght). Then $AdS_{5} \times S^{5}$ is not defining 5-dimensional quantum gravity by means of a SCFT.
All the known consistent examples of holography are constructed as the near horizon geometry of some piles of branes in string theory. The resulting bulk geometries are always of the form $AdS_{d} \times (something)$, always with ten or eleven macroscopic dimensions and never of the form $AdS_{d}$ (such as the case of pure gravity in 3d) or $AdS_{d} \times K$ where $K$ is some compact geometry as needed to describe 4d quantum gravity.
In that sense, $AdS/CFT$ is not a useful phenomenological scenario as the usual string theory compactifications are. A parametric separation between the KK-scale and the Hubble scale is the most basic requirement for phenomenology that is abscent in holographic setups.
Update: I recently learned that the path integral of three dimensional Chern-Simons theory can be viewed (under some hypothesis) as equivalent to $\mathcal{N}$=4 SYM in four dimensions. Reference: A new look at the path integral in quantum mechanics.
