Let $\Gamma$ be either the infinite cyclic group $\mathbb{Z}$ or the
Baumslag-Solitar group $\mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$. Let $K$ be
a slice knot admitting a slice disc $D$ in the 4-ball whose exterior has
fundamental group $\Gamma$. We classify the $\Gamma$-homotopy ribbon slice
discs for $K$ up to topological ambient isotopy rel. boundary using surgery
theory. In the infinite cyclic case, there is a unique equivalence class of
such slice discs. When $\Gamma$ is the Baumslag-Solitar group, there is at most
one $\Gamma$-homotopy ribbon slice disc for each lagrangian of the Blanchfield
pairing of $K$.