A finitely presented group with unbounded dead-end depth

The dead-end depth of an element $g$ of a group $G$, with respect to a generating set $\mathcal{A}$, is the distance from $g$ to the complement of the radius $d_{\mathcal{A}}(1,g)$ closed ball, in the word metric $d_{\mathcal{A}}$ defined with respect to $\mathcal{A}$. We exhibit a finitely presented group $G$ with a finite generating set with respect to which there is no upper bound on the dead-end depth of elements.