The hyperbolic conjecture on polynomial independence for point configurations

Let $(x_1,\dotsc,x_n)\subset H^3$ be $n$ distinct ordered points. For $i\ne j$, let $u_{ij}$ be the point where the oriented geodesic from $x_i$ to $x_j$ meets the sphere at infinity $S^2\cong\mathbb{CP}^1$, and define $\beta_i$ by

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

Hyperbolic conjecture. For every such configuration, the polynomials $\beta_1(z),\dotsc,\beta_n(z)$ are linearly independent over $\mathbb{C}$. This is presented as a stronger version of the Euclidean conjecture, using the ideal boundary of hyperbolic three-space; the supplied text gives no evidence that it has been resolved.