Is the gauge/gravity (or AdS/CFT) duality believed to be exact? I was wondering about the implications of the gauge/gravity (or AdS/CFT in a more restrictive sense) duality for the way we deal with physical theories, and I was wondering if the duality was believed to be exact, i.e. that the gauge theory and the gravity counterpart are describing exactly the same situation?
If the answer is yes, then as a follow-up question, does that mean that we would not be able to say how many dimensions reality has, in case we were to find an unified gauge theory in $d$ dimensions dual with a gravity theory in $d+1$ dimensions? We would also then not be able to differentiate if an effect is due to gravity or to gauge theory (for example in the case we were to find a gravity dual to QCD, although this is not a CFT)?
Or is it that the extra degree of freedom induced by the $+1$ dimension on the gravity side is somehow encoded in the gauge group?
Thank you in advance for your answers.
Edit: Funnily enough, I just came across this statement in the lecture notes by Năstase, which I guess answers the first question (the 2nd one remains):

 A: Re second question: we can formulate quantum mechanics in coordinate representation or equivalently in momentum representation. So do we live in position space or in momentum space? The answer is that it doesn't matter. Some questions are easier to answer in position space, some in momentum space. Everyday life is more easily addressed in position space.
The same is with AdS/CFT: you can describe physics in terms of gravity theory or in terms of a CFT. Some questions are easier to answer in the former formulation, some are easier in the latter. Most experiments are easier to describe in gravity formulation.

Or is it that the extra degree of freedom induced by the +1 dimension on the gravity side is somehow encoded in the gauge group?

I am not sure what you mean by that. The statement of AdS/CFT is that the gravity theory is equivalent to the CFT. This means that you can (in principle) describe the same experiment in gravity theory or in CFT. You don't use these two descriptions simultaneously.
