Converse characterization of sign patterns with unique sepr-sequences

Let $\mathcal P$ be an $n\times n$ sign pattern. For each $k=1,\ldots,n$, let $t_k(\mathcal P)$ be the $k$th term of its sepr-sequence, and say that a $k\times k$ principal subpattern has a signed determinant when its determinant sign is fixed over all real matrices realizing that sign pattern.

Converse characterization conjecture. The following are equivalent:

1. $\mathcal P$ has a unique sepr-sequence.
2. For each $k=1,\ldots,n$, either every $k\times k$ principal subpattern has a signed determinant, or $t_k(\mathcal P)={\tt S}^{*}$ and there are three $k\times k$ principal subpatterns whose signed determinants are respectively $+$, $-$, and $0$.

This conjecture is stated as the converse of the paper's sufficient-condition result for unique sepr-sequences. It gives a proposed complete characterization, but the supplied source provides no evidence that it has been resolved.