The sharp Lipschitz bound for Möbius transformations of the Hilbert metric in the unit ball

Let $\mathbb{B}^n$ be the unit ball, let $h_{\mathbb{B}^n}$ denote its Hilbert metric, and let $T_a$ be the transformation defined in the paper for $a\in\mathbb{B}^n$. For $x,y,a\in\mathbb{B}^n$, with $x\ne y$, consider the ratio of the distances after and before applying $T_a$.

The conjectured Lipschitz bound. For all $x,y,a\in\mathbb{B}^n$,

$$\frac{h_{\mathbb{B}^n}(T_a(x),T_a(y))}{h_{\mathbb{B}^n}(x,y)}\leq 1+|a|.$$

Numerical tests suggest that this bound holds. It would give a uniform Lipschitz estimate for the transformations $T_a$ in the Hilbert metric, with dependence on the parameter $a$ only through its Euclidean norm; no proof or resolution is supplied here.