Let $\mathbb{H}^A$ and $\mathbb{H}^B$ be finite dimensional Hilbert spaces. Consider the set $S$ of all bipartite density operators $\rho \in D(\mathbb{H}^A \otimes \mathbb{H}^B)$ such that every eigenspace of $\rho$ has an orthonormal basis consisting of separable vectors, i.e. vectors of the form $| e_j \rangle \otimes | f_k \rangle$ for some $| e_j \rangle \in \mathbb{H}^A$ and $| f_k \rangle \in \mathbb{H}^B$. Are there any equivalent characterizations or useful properties of this set $S$?
For instance, every $\rho \in S$ can be written as
$$\rho = \sum_{j,k} p_{j,k} | e_j \rangle \otimes | f_k \rangle \langle e_j | \otimes \langle f_k |,$$
where the $p_{j,k}$ are probabilities and the $| e_j \rangle \otimes | f_k \rangle$ are the separable orthonormal eigenvectors of $\rho$. Then every $\rho \in S$ is separable. However, from this question, it appears that not every separable state belongs to $S$. Can we characterize which separable states belong to $S$?
This question seems to be related.
