  We exhibit infinitely many ribbon knots, each of which bounds infinitely many
pairwise non-isotopic ribbon disks whose exteriors are diffeomorphic. This
family provides a positive answer to a stronger version of an old question of
Hitt and Sumners. The examples arise from our main result: a classification of
fibered, homotopy-ribbon disks for each generalized square knot $T_{p,q}\#
\overline{T}_{p,q}$ up to isotopy. Precisely, we show that each generalized
square knot bounds infinitely many pairwise non-isotopic fibered,
homotopy-ribbon disks, all of whose exteriors are diffeomorphic. When $q=2$, we
prove further that infinitely many of these disks are also ribbon; whether the
disks are always ribbon is an open problem.
