Ng's conjecture relating abelian contact homology to the branched-cover character ring

Let $K$ be a knot in $bS^3$, let $bbSigmad_2K$ be its 2-fold branched covering, and let $X(bbSigmad_2K)$ be the $bSL_2(bC)$ character variety of $bpi_1(bbSigmad_2K)$. Write $a_{ij}$ for the generators of the degree-zero abelian contact homology algebra $HC_0^{ab}(K)$ and $z_{ij}=\operatorname{tr}(\rho(m_im_j))$ for the corresponding coordinate functions on $X(bbSigmad_2K)$. Ng's conjecture. For any knot $K$, the homomorphism

$$g: HC_0^{ab}(K) \otimes bC \longrightarrow bC[X(bbSigmad_2K)]$$

defined by $g(a_{ij})=-z_{ij}$ $(1\leq i<j\leq n)$ and $g(1)=1$ gives an isomorphism. In particular, $HC_0^{ab}(K)\otimesbC$ and $bC[X(bbSigmad_2K)]$ are isomorphic. This conjecture proposes that the abelianized degree-zero contact homology detects the coordinate ring of the character variety of the 2-fold branched cover for every knot; its resolution is not supplied here.