Metricity threshold for the point pair function outside the unit ball

Let $\mathbb{B}^n$ be the unit ball in $\mathbb{R}^n$, and let $p^\alpha_{\mathbb{R}^n\backslash\overline{\mathbb{B}}^n}$ be the function

$$p^\alpha_{\mathbb{R}^n\backslash\overline{\mathbb{B}}^n}(x,y)=\frac{|x-y|}{\sqrt{|x-y|^2+\alpha(|x|-1)(|y|-1)}}.$$

Metricity conjecture. For a constant $\alpha>0$, this function, defined for $x,y\in\mathbb{R}^n\backslash\overline{\mathbb{B}}^n$ and $n\geq2$, is a metric if and only if $\alpha\leq12$. The threshold $12$ is proposed on the basis of numerical tests; its validity remains open.