The four-point angular formula conjecture for the Atiyah-Sutcliffe determinant

Let $v_{ab}\in S^2$ be the unit vectors associated with a configuration of four distinct points in $\mathbb{R}^3$, let $D$ be the normalized Atiyah-Sutcliffe determinant, and let $\operatorname{Av}$ denote averaging over the symmetric group $\Sigma_4$. Four-point angular formula conjecture. If $n=4$, then

$$\operatorname{Re}(D)=\frac{3}{8}+\frac{3}{2}\operatorname{Av}((v_{12},v_{13}))+\frac{3}{2}\operatorname{Av}((v_{12},v_{13})(v_{14},v_{24}))+\cdots+\frac{3}{8}\operatorname{Av}((v_{12},v_{34})(v_{13},v_{24}))+\frac{1}{2}\operatorname{Av}((v_{12},v_{14})(v_{13},v_{23})(v_{24},v_{34})),$$

and

$$\operatorname{Im}(D)=-\frac{1}{32}\operatorname{Av}(\det(v_{12},v_{13},v_{24}))-\frac{1}{32}\operatorname{Av}(\det(v_{12},v_{14},v_{23})(v_{24},v_{34})).$$

The formula is presented as an application of the preceding angular-factor conjecture, but the supplied text does not establish it or specify its resolution status.