Lipschitz conjecture for Möbius transformations and the generalized point pair function

Let $T_a:\mathbb{B}^2\to\mathbb{B}^2$ be the Möbius transformation

$$T_a(z)=\frac{z-a}{1-\overline{a}z}.$$

For $x,y,a\in\mathbb{B}^2$ and $\alpha>0$, the generalized point pair function conjecture.

$$\frac{1}{1+|a|}p^\alpha_{\mathbb{B}^2}(x,y)\leq p^\alpha_{\mathbb{B}^2}(T_a(x),T_a(y))\leq(1+|a|)p^\alpha_{\mathbb{B}^2}(x,y).$$

Computer tests suggest that the Lipschitz constant of $T_a$ for the generalized point pair function is $1+|a|$, independently of $\alpha>0$, extending analogous conjectures for several intrinsic metrics and quasi-metrics on the unit disk.