Character-variety conjecture for the complexified cord ring

Let $K$ be a knot, let $\Sigma_2(K)$ be its double branched cover, and let $HC_0^{\operatorname{ab}}(K)$ denote the abelianized degree-zero contact homology, identified with the cord ring. Consider the map from the complexified cord ring to the coordinate ring of the $SL_2(\mathbb{C})$ character variety of $\Sigma_2(K)$ constructed in Proposition [the source's Proposition~]. Character-variety conjecture. This map is always an isomorphism, so that $HC_0^{\operatorname{ab}}(K)\otimes\mathbb{C}$ is precisely the coordinate ring of the character variety of $\Sigma_2(K)$. The claim is known for two-bridge knots in the discussion immediately preceding it; the general case is proposed on the basis of computations with small knots.