Dominance of the complete-structure sector in the Turaev–Viro sum

Let $K$ be a hyperbolic knot, let $U$ be the neighborhood of $s=1$ appearing in the parametrized colored-Jones conjecture, and set $s=M/(N+1/2)$. For a sequence $a_N$, write $a_N=o(b_N)$ when $\lim_{N\to\infty}a_N/b_N=0$.

Complete-structure dominance conjecture. The colored Jones terms with $s\in U$ dominate those with $s\notin U$:

$$\sum_{M:s\notin U}\left|J_M\left(K,e^{\frac{2\pi i}{N+1/2}}\right)\right|^2=o\left(\sum_{M:s\in U}\left|J_M\left(K,e^{\frac{2\pi i}{N+1/2}}\right)\right|^2\right).$$

This is intended to express that contributions away from the complete hyperbolic structure are exponentially smaller and may be neglected in the Turaev–Viro asymptotics. The supplied text gives motivation from Thurston's volume comparison but no proof for general hyperbolic knots.