The Minkowski space conjecture on celestial polynomial independence

Let $\xi_1,\dotsc,\xi_n\subset\mathbb{R}^{3,1}$ be $n$ non-intersecting world lines in Minkowski space, and let $x_i\in\xi_i$ be distinct events. For $i\ne j$, let $u_{ij}$ be the point on the celestial sphere $S^2\cong\mathbb{CP}^1$ where the past light cone at $x_i$ intersects $\xi_j$, and define $\beta_i$ from the roots $u_{ij}$ as above. Minkowski space conjecture. The polynomials $\beta_1(z),\dotsc,\beta_n(z)$ are linearly independent over $\mathbb{C}$. This is the strongest of the three geometric versions, formulated for moving stars and their celestial observations in flat spacetime; the supplied text does not indicate whether it is known or open.