  Given a configuration $\mathbf{x}$ of $n$ distinct points in hyperbolic
$3$-space $H^3$, Michael Atiyah associated $n$ polynomials $p_1,\ldots,p_n$ of
a variable $t \in \mathbb{C}P^1$, of degree $n-1$, and conjectured that they
are linearly independent over $\mathbb{C}$, no matter which configuration
$\mathbf{x}$ one starts with. We prove this conjecture for $n=4$ in two cases:
in case the $4$ points are non-coplanar, and in case one of the points lies in
the hyperbolic convex hull of the other three.
