The chordal Lipschitz constant conjecture for Möbius transformations

Let $\mathbb{B}^n$ be the unit ball, let $\overline{\mathbb{R}}^n$ be the one-point compactification of $\mathbb{R}^n$, and let $q$ denote the chordal metric. For $a\in\mathbb{B}^n$, let $T_a$ be the Möbius transformation considered in the paper. Chordal Lipschitz constant conjecture. The Lipschitz constant of $T_a$ in the chordal metric is

$$\operatorname{Lip}(T_a|\overline{\mathbb{R}}^n)\equiv \sup\left\{\frac{q(T_a(x),T_a(y))}{q(x,y)}\,:\,x,y\in\overline{\mathbb{R}}^n\right\}=\frac{1+|a|}{1-|a|}.$$

The claim concerns the distortion of the hyperbolic isometry $T_a$ when measured with the chordal metric rather than the hyperbolic metric. The source presents it as suggested by computer tests, and no resolution is supplied there.