To each closed $3$-manifold $N$, there is a corresponding Witten--Reshetikhin--Turaev invariant $Z_k(N)$ depending on an integer $k$ (the level) and a Lie group $G$ (and perhaps we'll just concentrate on $G=SU(2)$ for the moment). Based on the perturbation analysis of the path integral "definition" of $Z_k(N)$, one predicts very precise asymptotics as $k\to\infty$, usually involving a sum/integral over the space of representations of $\pi_1(N)$ to $SU(2)$ (for instance, equation (1.32) in this paper: http://www.ams.org/mathscinet-getitem?mr=1133261)

Are any of these asymptotics rigorously proven? (the fact that the Volume Conjecture is still open leads me to think that the answer is "no")