The *asymptotic expansion conjecture* (AEC) states the following:

Let $M$ be a 3-manifold.
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 the 3-manfold invariant $Z_{k}^{G}(M)$ for large $r$ would be of the form
$$\sum_{j=0}^{n}e^{2\pi i\,r\,q_{j}}r^{d_{j}}b_{j}
(1+\sum_{l=1}^{\infty}a_{j}^{l}r^{-l}),$$
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 **against** 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".

*References:*

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

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