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.

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.

The asymptotic expansion conjecture (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 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.

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.