The metric conjecture for the function $w_{\mathbb{B}^n}$

Let $\mathbb{B}^n$ be the unit ball in $\mathbb{R}^n$. For $x,y\in\mathbb{B}^n\setminus\{0\}$, define $\widetilde{x}=x(2-|x|)/|x|$ and $\widetilde{y}=y(2-|y|)/|y|$, and set

$$w_{\mathbb{B}^n}(x,y)=\frac{|x-y|}{\min\{|x-\widetilde{y}|,|y-\widetilde{x}|\}}.$$

Also define $w_{\mathbb{B}^n}(x,0)=|x|/(2-|x|)$. The metric conjecture for the function $w_{\mathbb{B}^n}$. The function $w_{\mathbb{B}^n}$ is a metric on the unit ball. Numerical tests suggest that the triangle inequality holds, whereas the preceding results establish only that this function is a quasi-metric; whether it is a metric remains open.