Generic uniqueness of area minimizing disks for extreme curves

We show that for a generic nullhomotopic simple closed curve $\Gamma$ in the boundary of a compact, orientable, mean convex $3$-manifold $M$ with $H_2(M,{\bf Z})=0$, there is a unique area minimizing disk $D$ embedded in $M$ with $\partial D = \Gamma$. We also show that the same is true for nullhomologous curves in the absolutely area minimizing surface case.