The weak Atiyah–Sutcliffe conjecture for configurations in three-dimensional space

Let $\operatorname{Conf}_n(\mathbb{R}^3)$ be the configuration space of $n$ distinct points in $\mathbb{R}^3$. For a configuration $(p_1,\dots,p_n)$, let $\ell_i(p_1,\dots,p_n)$ be the complex line obtained from the symmetric product of the directions from $p_i$ to the other points. Atiyah's weak conjecture. The lines $\ell_1(p_1,\dots,p_n),\dots,\ell_n(p_1,\dots,p_n)$ are projectively independent for every $(p_1,\dots,p_n)\in\operatorname{Conf}_n(\mathbb{R}^3)$. This is equivalent to the nonvanishing of the associated determinant; it is known for $n\leq4$ and for configurations of some special types, but remains open in general.