Howson property and one-relator groups

We provide an example of a torsion-free word hyperbolic one-relator group G which does not have the Howson property (the finitely generated intersection property). Moreover, this group G contains a free subgroup of rank two that is not quasiconvex in G. The group G also turns out to be not subgroup separable. Similarities and differences with 3-manifold groups are discussed.