The geometric-solution conjecture for the complex volume of hyperbolic knots

Let $K$ be a hyperbolic knot, and let $x$ be an algebraic solution of the periodicity equations associated with the braid-induced triangulation of $S^3\setminus K$. For the $j$th octahedron, let $[\overset{k_j\phantom{\varepsilon}}{\mathsf{R}^{\varepsilon_j}}]$ denote its corresponding data, let $\boldsymbol{x}[j]$ denote the associated variables, and let $\operatorname{L}([\overset{k_j\phantom{\varepsilon}}{\mathsf{R}^{\varepsilon_j}}];\boldsymbol{x}[j])$ be the dilogarithm function defined in the paper. The complex volume of $K$ is $\mathrm{i}(\operatorname{Vol}(S^3\setminus K)+\mathrm{i}\operatorname{CS}(S^3\setminus K))$. Complex-volume conjecture. There exists an algebraic solution of the periodicity equations such that

$$\mathrm{i}\left(\operatorname{Vol}(S^3\setminus K)+\mathrm{i}\operatorname{CS}(S^3\setminus K)\right)=\sum_{j=1}^m\operatorname{L}([\overset{k_j\phantom{\varepsilon}}{\mathsf{R}^{\varepsilon_j}}];\boldsymbol{x}[j]).$$

The expected solution is geometric: the additional two points have canceling neighbors and determine a complete hyperbolic structure on $S^3\setminus K$. Numerical examples are discussed, but the authors state that they do not know how to extract such a preferred solution in general from the periodicity equations.