Suppression conjecture for colored Jones polynomials of Whitehead doubles

Let $K$ be a nontrivial knot, let $N$ be a positive integer, and let $n$ satisfy $\left|n-\frac{N}{2}\right|<N^\delta$ for some $\frac{1}{2}<\delta<\frac{2}{3}$. Write $\hat{J}_{K,2n+1}$ for the polynomial used in the paper, and evaluate it and its derivative at $e^{\frac{2\pi\sqrt{-1}}{N}}$. Suppression conjecture. For every nontrivial knot $K$,

$$\frac{\hat{J}_{K,2n+1}\left(e^{\frac{2\pi\sqrt{-1}}{N}}\right)}{t\frac{d}{dt}\hat{J}_{K,2n+1}\left(e^{\frac{2\pi\sqrt{-1}}{N}}\right)}=o(N^{-2})$$

uniformly on $\left|n-\frac{N}{2}\right|<N^\delta$ for some $\frac{1}{2}<\delta<\frac{2}{3}$. The conjecture formalizes the expected suppression of the undifferentiated term in the satellite-knot summation; the paper proves the relevant estimates only for the torus-knot family under consideration, so the assertion for general nontrivial knots remains open.