The angular-factor conjecture for the Atiyah-Sutcliffe determinant

Let $C_n(\mathbb{R}^3)$ be the configuration space of $n$ distinct points, and for $a\ne b$ let $v_{ab}=(\mathbf{x}_b-\mathbf{x}_a)/\|\mathbf{x}_b-\mathbf{x}_a\|\in S^2$. Write $(u,v)$ for the Euclidean inner product and $\det(u,v,w)$ for the scalar triple product. Angular-factor conjecture. The normalized Atiyah-Sutcliffe determinant $D$ can be expressed as a rational linear combination of terms whose factors are only of the two types

$$(v_{ab},v_{cd})\quad\text{and}\quad\det(v_{ab},v_{cd},v_{ef}),$$

where the indices range as in the definition of the vectors $v_{ab}$. Such an expression would make the rotational invariance of the determinant manifest; the text illustrates the conjecture with a partially displayed formula for $n=4$, but does not establish the conjecture or give a resolution status.