A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres (Reference for the geometry of horospheres), namely exponential decay of the projection onto a horosphere.

My question is: does a similar exponential decay still holds in a Gromov hyperbolic space ?

The precise statement would be something like: Let X be a $\delta$-hyperbolic space, $B$ a horoball and $H$ the corresponding horosphere. Assume that nearest point projection on $H$ is well defined and denote by $\pi$ this projection.

There exists $\alpha>0$ such that the following holds. Assume that $\gamma$ is a path in $X\setminus B$ such that $d(\gamma(t),H)\geq k>0$ for some $\alpha$ and for every $t$. Then, $l(\pi \circ \gamma)\leq e^{-\alpha k}l(\gamma)$, where $l$ is the length of a path.

Is there a reference for such a statement or something similar?

EDIT: Maybe the assumption that nearest point projection is too restrictive. A slightly weaker (but more general) statement would be:

Without assuming nearest point projection is well-defined but with the same notations, there exists $\alpha>0$ such that the following holds. Let $x,y\in X$ be such that there exists a path $\gamma$ from $x$ to $y$ such that for all $t$, $d(\gamma(t),H)\geq k>0$. Assume that $x$ and $y$ (coarsely) project onto $p$ and $q$ on the horosphere $H$. Then, $d(x,y)\geq e^{\alpha k}d(p,q)$.

(Maybe you should also require that $p$ and $q$ are far enough from each other, i.e. $d(p,q)\geq a$, where $a$ only depends on the hyperbolic parameters of $X$).