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 $j \to \infty$.

Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?

• The case where $n = 2$ and the $M_j$ have non-empty branch set might already be interesting, but I am most interested in examples of higher dimension.

• One can construct branched minimal immersions in $\mathbf{R}^{n+1}$ that converge to $2 \lvert B_1^n \times \{ 0 \} \rvert$ by solving a Dirichlet problem. But these have a boundary, so it's not clear how to adapt this to the sphere.