Conjectured sharp quasiinvariance of the distance-ratio metric under ball automorphisms

Let $a\in\mathbb{B}^n$ and let $h:\mathbb{B}^n\to\mathbb{B}^n$ be a Möbius transformation of the unit ball onto itself satisfying $h(a)=0$. The distance-ratio metric is denoted by $j_{\mathbb{B}^n}$. Distance-ratio metric conjecture.

$$\sup_{x,y\in\mathbb{B}^n\atop{x\neq y}}\frac{j_{\mathbb{B}^n}\left(h(x),h(y)\right)}{j_{\mathbb{B}^n}(x,y)}=1+|a|.$$

This conjecture seeks a sharp analogue for the distance-ratio metric of the proven quasiinvariance theorem for the quasihyperbolic metric under Möbius self-mappings of the unit ball.