  We answer a question of Livingston from 1982 by producing Seifert surfaces of
the same genus for a knot in $S^3$ that do not become isotopic when their
interiors are pushed into $B^4$. In particular, we identify examples where the
surfaces are not even topologically isotopic in $B^4$, examples that are
topologically but not smoothly isotopic, and examples of infinite families of
surfaces that are distinct only up to isotopy rel. boundary. Our main proofs
distinguish surfaces using the cobordism maps on Khovanov homology, and our
calculations demonstrate the stability and computability of these maps under
certain satellite operations.
