Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold $\mathcal{M}\subset {\mathbb{R}}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{h,\beta }$ and an oriented, closed submanifold $\mathrm{\Gamma }\subset \mathcal{M}$ of dimension $m-1$, which is a boundary in integral homology, we construct a complete metric space $\mathcal{B}$ of $C^{h,\alpha }$-perturbations of $\mathrm{\Gamma }$ inside $\mathcal{M}$, with $\alpha <\beta$, enjoying the following property. For the typical element $b\in \mathcal{B}$, in the sense of Baire categories, there exists a unique $m$-dimensional integral current in $\mathcal{M}$ which solves the corresponding Plateau problem and it has multiplicity one.