  If $\Gamma$ is the fundamental group of a complete finite volume hyperbolic
$3$-manifold, Guilloux conjectured that the Borel function on the
$\text{PSL}(n,\mathbb{C})$-character variety of $\Gamma$ should be rigid at
infinity, that is it should stay bounded away from its maximum at ideal points.
In this paper we prove Guilloux's conjecture in the particular case of the
reflection group associated to a regular ideal tetrahedron of $\mathbb{H}^3$.
