  In the topological category, the classification of homotopy ribbon discs is
known when the fundamental group $G$ of the exterior is $\mathbb{Z}$ and the
Baumslag-Solitar group $BS(1,2)$. We prove that if a group $G$ is geometrically
$2$-dimensional and satisfies the Farrell-Jones conjecture, then a condition
involving the fundamental group ensures that exteriors of aspherical homotopy
ribbon discs with fundamental group $G$ are s-cobordant rel.\ boundary. When
$G$ is good, this leads to the classification of such discs. As an application,
for any knot $J \subset S^3$ whose knot group $G(J)$ is good, we classify the
homotopy ribbon discs for $J \# -J$ whose complement has group $G(J)$. A
similar application is obtained for $BS(m,n)$ when $|m-n|=1$.
