Howson groups which are not strongly Howson

A group $G$ is called a Howson group if the intersection $H\cap K$ of any two finitely generated subgroups $H,K<G$ is again finitely generated, and called a strongly Howson group when a uniform bound for the rank of $H\cap K$ can be obtained from the ranks of $H$ and $K$. Clearly, every strongly Howson group is a Howson group, but it is unclear in the literature whether the converse is true. In this note, we show that the converse is not true by constructing the first Howson groups which are not strongly Howson.