A finite rewriting system is presented that does not satisfy the homotopical finiteness condition FDT, although it satisfies the homological finiteness condition FHT. This system is obtained from a group $G$ and a finitely generated subgroup $H$ of $G$ through a monoid extension that is almost an HNN extension. The FHT property of the extension is closely related to the ${\rm FP}_2$ property for the subgroup $H$, while the FDT property of the extension is related to the finite presentability of $H$. The example system separating the FDT property from the FHT property is then obtained by applying this construction to an example group.