Consider a geodesic, hyperbolic metric space $(X,d)$. For $x,y \in X$, consider $D(x,y)$ the infimum, over all $1$-Lipschitz paths $p:[a,b] \rightarrow X$ from $x$ to $y$, of the integral $\int_a^b e^{-(x|y)_{p(t)}}dt$, where $(x|y)_{p(t)}$ is the Gromov product.

Is the function $D(x,y)$ metrically proper ? More precisely, does there exist constants $A,B > 0$ such that $D(x,y) \geq Ad(x,y)-B$ ? Has this quantity already been studied ?