The conjectured exact value of the Möbius triangular-ratio Lipschitz constant

For $a\in (0,1)$, let $C(a)$ be the class of all Möbius transformations $h:\mathbb{B}^n\rightarrow\mathbb{B}^n$ satisfying $|h(0)|=a$, and define

$$L(a)=\sup\left\{\frac{s_{\mathbb{B}^n}(h(x),h(y))}{s_{\mathbb{B}^n}(x,y)}:x,y\in\mathbb{B}^n,\ x\neq y,\ h\in C(a)\right\}.$$

The conjectured value of $L(a)$. For $n=2$, numerical experiments suggest that

$$L(a)=1+a.$$

The preceding theorem proves the lower bound $L(a)\geq 1+a$ in dimension two, so the conjecture concerns the matching upper bound and would determine the optimal constant in the corresponding triangular-ratio metric estimate for conformal self-maps of the disk.