Conjectured Lipschitz bound for Möbius transformations of the unit disk in the t-metric

Let $\mathbb{B}^2=\{z\in\mathbb{C}:|z|<1\}$ be the unit disk, and for $a\in\mathbb{B}^2$ define the Möbius transformation

$$T_a:\mathbb{B}^2\to\mathbb{B}^2,\qquad T_a(z)=\frac{z-a}{1-\overline{a}z}.$$

Here $t_{\mathbb{B}^2}$ denotes the $t$-metric on $\mathbb{B}^2$. Möbius t-metric conjecture. For all $a,x,y\in\mathbb{B}^2$,

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

The general theorem gives the bound $2$ for conformal self-mappings, while this conjecture predicts the sharper constant $1+|a|$ for these particular Möbius transformations. It is supported by several numerical tests, but remains open in the source.