Which embedded minimal hypersurfaces $\Sigma^n$ in the unit sphere $S^{n+1}$ arise as limits of branched minimal immersions $M_j^n$, in the sense that $M_j \to 2 \Sigma$ as currents (or varifolds)?

By 'branched minimal immersion' I mean a stationary integral varifold $V$ whose singularities are either immersed or branch points. (I believe this can be expressed also in terms of smooth maps into $S^{n+1}$ which fail to be immersed at the pre-image of the branch set, but the translation might be a bit awkward.)

I am only interested in those surfaces that are genuinely branched, at least in the sense that $M_j$ cannot be decomposed into the union of two minimal surfaces $M_j^1$ and $M_j^2$ with $M_j^i \to \Sigma$ as $j \to \infty$.