Generic bumpy boundary conjecture for minimal submanifolds

Let $G$ be a finite set of isometries of $\mathbf{R}^n$. A smooth, closed, $G$-invariant, embedded submanifold of $\mathbf{R}^n$ is one whose boundary-bounded minimal submanifolds are considered below. Generic bumpy boundary conjecture. For a generic smooth, closed, $G$-invariant, embedded submanifold of $\mathbf{R}^n$, none of the smooth immersed minimal submanifolds it bounds have nontrivial Jacobi fields that vanish at the boundary. The conjecture is motivated by the bumpy metrics theorem for closed minimal submanifolds and by the analogous genericity phenomenon for minimal surfaces with boundary; its general validity is posed as an open problem.