The <b><i>asymptotic expansion conjecture</i></b> (AEC) states the following:

Putting $r := k+h^{\vee}$ with
$h^{\vee}$ the dual Coxeter number of the Lie algebra of $G$, the
AEC states that the asymptotic expansion
of $Z_{k}^{G}(M)$ for large $r$ would be of the form
$$\sum_{j=0}^{n}\exp(2\pi\sqrt{-1}rq_{j})r^{d_{j}}b_{j}
(1+\sum_{l=1}^{\infty}a_{j}^{l}r^{-l}),$$
where
$d_{j}\in\Bbb{Q}$, $b_{j},a_{j}^{l}\in\Bbb{C}$, and
$q_{j}\in\Bbb{R}/\Bbb{Z}$. Moreover, the set $\{q_{0}=0,
q_{1},\dots,q_{n}\}$ should consist of the values of the Chern-Simons
functional.

See the paper [AH06] for a survey of known results on the AEC (ok, it's 6 years old...).
Note that, according to [AH06], the paper [KSV97] suggests numerical 
evidence <b>against</b> the conjectures for the 3-manifold $S^3 (4_1 ( 
−n/1))$, $n = 7, 16, 22$ (a manifold obtained by doing a particular Dehn surgery on a figure-eight knot), demonstrating a 
contribution from a non-Chern–Simons-value phase of order 
$−2$ in the level.

<br><br><i>References:</i><br>
[AH06] Andersen, Jørgen Ellegaard; Hansen, Søren Kold
<i>Asymptotics of the quantum invariants for surgeries on the figure 8 knot.</i> 
J. Knot Theory Ramifications 15 (2006), no. 4, 479–548. 

[KSV97] Michael Karowski, Robert Schrader, and Elmar Vogt. <i>Invariants of three-manifolds, unitary representations of the mapping class group, and numerical calculations.</i> Experiment. Math., 6(4):317–352, 1997.