Property $P_{naive}$ for acylindrically hyperbolic groups

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1,\dots ,h_k$, there exists another element $\gamma \mathrm{\ne }1$ such that for all $i$, $\langle h_i, \gamma \rangle = \langle h_i \rangle* \langle \gamma \rangle$. We also obtain that if a collection of subgroups $H_1,\dots ,H_k$ is a hyperbolically embedded collection, then there is $\gamma \mathrm{\ne }1$ such that for all $i$, $\langle H_i, \gamma \rangle = H_i * \langle \gamma \rangle$.