Consider the following theorem, proved in 
[this paper][1]:


> **Theorem**. Suppose we have a sequence  $(\Sigma_j, \partial \Sigma_j) \subset (M, \partial M)$ of immersed free boundary minimal $k$-dimensional submanifolds, where $1 \leq k \leq n$, with uniformly bounded area and second fundamental form. Then, after passing to a subsequence, $(\Sigma_j, \partial \Sigma_j)$ converges smoothly and locally uniformly to $(\Sigma, \partial \Sigma) \subset (M, \partial M)$, which is a smooth immersed free boundary minimal $k$-dimensional submanifold.


My questions are:

**Question 1**: If we assume additionally that all the submanifolds in the sequence are embedded, is it true that the limit surface is also embedded?

**Question 2**: If $M$ has dimension $3$ and 
  $(\Sigma_j, \partial \Sigma_j)$ is a sequence of compact, connected, oriented and properly embedded free boundary minimal surfaces that converges as in the theorem to $(\Sigma, \partial \Sigma)$, is it true that there exists $N \geq 1$ such that $ [\Sigma_j] = [\Sigma] \in H_2(M, \partial M; \mathbb{Z})$ for all $j \geq N$?

  [1]: https://arxiv.org/pdf/1611.02605.pdf