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Pure $2+1$-dimensional gravity in $AdS_3$ (parametrized as $S= \int d^3 x \frac{1}{16 \pi G} \sqrt{-g} (R+\frac{2}{l^2})$) is a topological field theory closely related to Chern-Simons theory, and at least naively seems like it may be renormalizable on-shell for certain values of $l/G$. This is a theory which has been studied by many authors, but I can't seem to find a consensus as to what the CFT dual is. Here's what I've gathered from a cursory literature search:

Witten (2007) suggests that the dual is the monster theory of Frenkel, Lepowsky, and Meurman for a certain value of $l/G$; his argument applies when the central charges $c_L$ and $c_R$ are both multiplies of $24$. In his argument, he assumes holomorphic factorization of the boundary CFT, which seems to be fairly controversial. His argument does produce approximately correct entropy for BTZ black holes, but a case can be made that black hole states shouldn't exist at all if the CFT is holomorphically factorized. He also gave a PiTP talk on the subject. Witten himself is unsure if this work is correct.

In a recent 2013 paper, McGough and H. Verlinde claim that "The edge states of 2+1-D gravity are described by Liouville theory", citing 5 papers to justify this claim. All of those are before Witten's 2007 work. Witten's work does mention Liouville theory, and has some discussion, but he doesn't seem to believe that this is the correct boundary theory, and Liouville theory is at any rate not compatible with holomorphic factorization. This paper also claims that "pure quantum gravity...is unlikely to give rise to a complete theory." Similar assertions are made in a few other papers.

Another proposal was made in Castro et.al (2011), relating this to minimal models such as the Ising model. Specifically, they claim that the partition function for the Ising model is equal to that of pure gravity $l=3G$, and make certain claims about higher spin cases.

It doesn't seem to me that all of these can simultaneously be true. There could be some way to mitigate the differences between the proposals, but my scan of the literature didn't point to anything. It seems to me that no one agrees on the correct theory. I'm not even sure if these are the only proposals, but they're the ones that I'm aware of.

First, are my above statements regarding the three proposals accurate? Also, is there any consensus in the majority of the HET community as to whether pure quantum gravity theories in $AdS_3$ exist, and if so what their CFT duals are? Finally, if there is no consensus, what are the necessary conditions for each of the proposals to be correct?

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Without reading your whole question and just answering the title:

I think that still is an (very interesting) open problem.

See e.g., Five Problems in Quantum Gravity - Andrew Strominger http://arxiv.org/abs/arXiv:0906.1313

On very general grounds [15], we expect that 3D AdS gravity should be dual to a 2D CFT with central charge c = 3l . 2G Solving the theory is equivalent to specifying this CFT. It was suggested in [23] that, rather than directly quantizing the Einstein- Hilbert action, this CFT might simply be deduced by various consistency requirements. Namely, the central charge must be c = 3l 2G , Z must be modular invariant (since these are large diffeomorphisms) and its pole structure must reflect the fact that there are no perturbative excitations. Adding the additional assumption of holomorphic factorization (i.e. decoupling of the left and right movers in the CFT), it was shown [23] that Z is uniquely determined to be a certain modular form Zext . Unfortunately Zext does not agree with the Euclidean sum-over-geometries [25] which indicates that the assumption is not valid for pure gravity.3 Modular invariance and the restriction on the pole structure are still strong, if not uniquely determining, hints on the form of Z for pure gravity. Determining Z for pure 3D quantum Einstein gravity - if it exists - is an important open problem.

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Be aware that i could easily be not be up to date. So take that with a grain of salt. Hopefully a more knowledgeable user will add something. –  ungerade Nov 1 '13 at 1:22
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Thanks. Two of the three papers I cited are more recent than 2009, but assuming it hasn't changed, this at least negatively answers the question "is there a consensus?". –  user32020 Nov 1 '13 at 1:23
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