Finite approximabiliyt of free products with respect to occurrence

We shall say that a group $G$ belongs to a class $\Phi\mathrm{AB}_\omega$ if and only if for any finitely generated subgroup $H$ of $G$ and any element $g$ of $G$ that does not lie in $H$ there exists a homomorphism of $G$ into a finite group such that the image of $g$ does not belong to the image of the subgroup $H$. We prove that the class $\Phi\mathrm{AB}_\omega$ is closed under the operation of free multiplication.