Like a lot of dualities and ideas in physics there is no proof for AdS/CFT and I don't see there being one for awhile. Its hard enough to make QFT rigorous by itself!
Also AdS/CFT is a little broad. There are a lot of dualities that go under AdS/CFT, the most famous one being what Juan Maldacena originally proposed using N=4 SYM. There are proposed dualities that aren't well understood (like AdS2/CFT1 duality). The ones that are accepted as true are generally ones that either have been proved to be equivalent perturbatively and/or ones where its been observed that all the correlation functions seem to match.
Experimentally there was some interesting work involving AdS/CFT with respect to perfect liquids and the ratio of shear viscosity to entropy and Subir Sachdev has been doing a lot of work connecting it to condensed matter physics. The problem there is that the field theories that are used in gauge gravity dualities are typically supersymmetric and conformal so an exact agreement isnt exactly likely.
There's also the fact that it can be incredibly useful so a lot theorists hope its right!