  Expanding on work by Conway, Orson, and Powell, we study the isotopy classes
rel. boundary of nonorientable, compact, locally flatly embedded surfaces in
$D^4$ with knot group $\mathbb{Z}_2$. In particular we show that if two such
surfaces have fixed knot boundary $K$ in $S^4$ such that $\vert \det(K) \vert
=1$, the same normal Euler number, and the same nonorientable genus $4$ or $5$,
then they are ambiently isotopic rel. boundary.
  This implies that closed, nonorientable, locally flatly embedded surfaces in
the $4$-sphere with knot group $\mathbb{Z}_2$ of nonorientable genus $4$ and
$5$ are topologically unknotted. The proof relies on calculations, implemented
in Sage, which imply that the modified surgery obstruction is elementary.
Furthermore we show that this method fails for nonorientable genus $6$ and $7$.
