The eventual monotonicity conjecture for colored Jones growth

For a knot $K$ in $S^3$, define the sequence

$$a_n=\frac{\log\left|J_K(n)\left(e^{\frac{2\pi i}{n}}\right)\right|}{n},$$

where $J_K(n)$ is the colored Jones polynomial. The eventual monotonicity conjecture. For every knot $K$ in $S^3$, the sequence $a_n$ is eventually decreasing and bounded above by zero. This conjecture would provide a useful clue toward proving existence of the limit in the Volume Conjecture, but the source gives no resolution.