A Gromov hyperbolic metric vs the hyperbolic and other related metrics

We mainly consider two metrics: a Gromov hyperbolic metric and a scale invariant Cassinian metric. A metric space $(D,d)$ is called Gromov hyperbolic if and only if there exist a constant $\beta>0$ such that
$$d(x,z)+d(y,w)\le (d(x,w)+d(y,z))\vee (d(x,y)+d(z,w))+2\beta $$
for all points $x,y,z,w \in D$.
For a domain $D\subsetneq \mathbb{R}^n$ equipped with the Euclidean metric, the $u_D$-metric [1] is defined by
$$ u_D(x,y)=2\log \frac{|x-y|+\max\{{\rm dist}(x,\partial D),{\rm dist}(y,\partial D)\}} {\sqrt{{\rm dist}(x,\partial D)\,{\rm dist}(y,\partial D)}}, \quad x,y\in D. $$
Ibragimov proved in [1] that the $u_D$-metric is Gromov hyperbolic and it coincides with the Vuorinen's distance ratio metric [3,4] in punctured spaces $\mathbb{R}^n\setminus\{p\}$, for $p\in\mathbb{R}^n$.
A scale invariant version of the Cassinian metric has been studied by Ibragimov in [2] which is defined by
$$ \tilde{\tau}_D(x,y)=\log\left(1+\sup_{p\in \partial D}\frac{|x-y|}{\sqrt{|x-p||p-y|}}\right), \quad x,y\in D\subsetneq \mathbb{R}^n. $$
The interesting part of this metric is that many properties in arbitrary domains are revealed in the setting of once-punctured spaces. For example, $\tilde{\tau}_D$ is a metric in an arbitrary domain $D\subsetneq \mathbb{R}^n$ if it is a metric on once-punctured spaces. The $\tilde{\tau}_D$-metric is comparable with the Vuorinen's distance ratio metric in arbitrary domains $D\subsetneq \mathbb{R}^n$ if they are comparable in the punctured spaces (see [2]).
Our purpose is to compare the $u_D$-metric with the hyperbolic and the $\tilde\tau$-metrics.
This is a joint work with Manas Ranjan Mohapatra.