Dimension-independent quasiconformal distortion conjecture for the hyperbolic distortion function

Let $K>1$, $n>2$, and $r\in(0,1)$. Define $\alpha=K^{1/(1-n)}$ and let $\varphi_{K,n}$ be the quasiconformal distortion function. Hyperbolic distortion conjecture.

$$\operatorname{arctanh}\varphi_{K,n}(\operatorname{th} r)\leq2\operatorname{arctanh}\left(\varphi_{K,2}\left(\operatorname{th}\frac12\right)\right)\max\{r,r^\alpha\}.$$

The conjecture would improve the paper's quasiconformal distortion theorem by replacing the dimension-dependent distortion expression with a bound involving the planar function $\varphi_{K,2}$; the supplied text does not indicate that it has been proved or disproved.