The abelian invariant–Alexander polynomial conjecture for links

Let $L\subset S^3$ be a link with $N$ components, let $\Delta_L(t_1,\ldots,t_N)$ be its Alexander polynomial, and let $\rho\colon \pi L\to \operatorname{SO}(3)$ be an abelian representation of the link group. The invariant $I_\rho(L)$ is defined for this representation. The abelian invariant–Alexander polynomial conjecture.

$$I_\rho(L)=\begin{cases} -\lvert\Delta_L(e^{i\varphi_1})\rvert^{-4}(2-2\cos\varphi_1)^2,&N=1,\\ -\lvert\Delta_L(e^{i\varphi_1},\ldots,e^{i\varphi_N})\rvert^{-4},&N>1. \end{cases}$$

This conjecture proposes that the abelian version of the Euclidean invariant is determined by the Alexander polynomial. The paper reports computations supporting the formula for other knots and links, but does not establish it in general.