Howson's property for semidirect products of semilattices by groups

An inverse semigroup $S$ is a Howson inverse semigroup if the intersection of finitely generated inverse subsemigroups of $S$ is finitely generated. Given a locally finite action $\theta$ of a group $G$ on a semilattice $E$, it is proved that $E{\ast }_{\theta }G$ is a Howson inverse semigroup if and only if $G$ is a Howson group. It is also shown that this equivalence fails for arbitrary actions.