The Euclidean conjecture on polynomial independence for point configurations

Let $(x_1,\dotsc,x_n)\subset\mathbb{R}^3$ be $n$ distinct points. For each $i$, define $u_{ij}=(x_j-x_i)/\lVert x_j-x_i\rVert\in S^2\cong\mathbb{CP}^1$ for $j\ne i$, and let

$$\beta_i(z)=\prod_{j\ne i}(z-u_{ij}).$$

Euclidean conjecture. For every such configuration, the polynomials $\beta_1(z),\dotsc,\beta_n(z)$ are linearly independent over $\mathbb{C}$. This is the first and simplest version of the conjecture, associating polynomials to configurations of points in Euclidean three-space; its resolution is not indicated in the supplied text.