Relations between classes of potentially stable sign patterns

Two subsets of the potentially stable sign patterns of order $n$ have recently been defined, namely, those that allow sets of (refined) inertias $\mathbb{S}_n$ and $\mathbb{H}_n$. For $n=2$ and $n=3$, it is proved that a sign pattern is potentially stable if and only if it is sign stable, allows $\mathbb{S}_n$, or allows $\mathbb{H}_n$. This result is also true for sign patterns of order $4$ with associated graph that is a tree, remains open for non-tree potentially stable sign patterns of order $4$, and is false for potentially stable sign patterns of orders greater than or equal to $5$.