FPRAS hardness conjecture for Jones polynomial evaluation

For an integer $ell$ with $\ell\geq 3$, let $q=e^{\pi i/\ell}$, and let $J_L(q^2)$ denote the Jones polynomial of a link $L$ evaluated at $q^2$. An FPRAS is a fully polynomial randomized approximation scheme, and $RP$ and $NP$ are the standard randomized and nondeterministic polynomial-time complexity classes.

Jones-polynomial FPRAS conjecture. There is no FPRAS for evaluating $J_L(q^2)$ except at the special points $\ell\in\{1,2,3,4,6\}$, provided $RP\neq NP$.

At the exceptional points the Jones polynomial degenerates to a classical link invariant computable in polynomial time, whereas exact evaluation is $\#P$-hard at the other roots of unity described in the paper. The conjecture concerns the stronger approximation-complexity question.