  We prove that for any countable acylidrically hyperbolic group $G$, there
exists a generating set $S$ of $G$ such that the corresponding Cayley graph
$\Gamma(G,S)$ is hyperbolic, $|\partial\Gamma(G,X)|>2$, the natural action of
$G$ on $\Gamma(G,S)$ is acylindrical, and the natural action of $G$ on the
Gromov boundary $\partial\Gamma(G,S)$ is hyperfinite. This result broadens a
class of groups that admit a non-elementary acylindrical action on a hyperbolic
space with hyperfinite boundary action.
