Approximation and exact computation conjecture for finite-group link homomorphism invariants

Let $G$ be a finite group, let $L$ be a link, and define

$$H_L(G)=\left|\operatorname{Hom}(\pi_1(S^3\setminus L),G)\right|.$$

An FPRAS is a fully polynomial randomized approximation scheme, and a group is solvable if it has a finite derived series terminating in the trivial group.

Finite-group link-invariant conjecture. (a) There exists an FPRAS for computing $H_L(G)$ for any finite group $G$. (b) If $G$ is solvable, then there is a polynomial-time algorithm for exact computation of $H_L(G)$.

The invariant arises from the modular category $\operatorname{Rep}(D G)$, where $D G$ is the Drinfeld double of $G$. Its computational complexity had not been studied in the source, so both the randomized approximation claim and the exact solvable-group claim were presented as conjectural.