  We construct an example of an isometric action of $F(a,b)$ on a
$\delta$-hyperbolic graph $Y$, such that this action is acylindrical, purely
loxodromic, has asymptotic translation lengths of nontrivial elements of
$F(a,b)$ separated away from $0$, has quasiconvex orbits in $Y$, but such that
the orbit map $F(a,b)\to Y$ is not a quasi-isometric embedding.
