Bodine et al.'s conjecture on irreducible sign patterns requiring $\mathbb{H}_n$

Let $n\ge 3$, let $\mathbb{H}_n=\{(0,n,0,0),(0,n-2,0,2),(2,n-2,0,0)\}$, and let an $n\times n$ sign pattern ${\cal A}$ require $\mathbb{H}_n$ when

$$\mathbb{H}_n=\{\operatorname{ri}(B)\mid B\in Q({\cal A})\}.$$

A sign pattern is irreducible if it is not permutation-similar to a block upper-triangular sign pattern with more than one diagonal block. Bodine et al.'s conjecture. No $n\times n$ irreducible sign pattern that requires $\mathbb{H}_n$ exists for $n$ sufficiently large, possibly $n\ge 8$. This conjecture concerns the existence of sign patterns whose qualitative classes realize exactly the refined inertias in $\mathbb{H}_n$; such refined inertias can signal the onset of periodic solutions by Hopf bifurcation. The source gives no resolution, so the conjecture remains open.