  Let $X$ be a closed indefinite $4$-manifold with $b_+(X) = 3 \; ({\rm mod} \;
4)$ and with non-vanishing mod $2$ Seiberg--Witten invariants. We prove a new
lower bound on the genus of a properly embedded surface in $X \setminus B^4$
representing a given homology class and with boundary a quasipositive knot $K
\subset S^3$. In the null-homologous case our inequality implies that the
minimal genus of such a surface is equal to the slice genus of $K$. If $X$ is
symplectic then our lower bound differs from the minimal genus by at most $1$
for any homology class that can be represented by a symplectic surface. Along
the way, we also prove an extension of the adjunction inequality for closed
$4$-manifolds to classes of negative self-intersection without requiring $X$ to
be of simple type.
