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The refined inertia of a square real matrix $A$ is the ordered $4$-tuple$(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number ofeigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is thenumber of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pureimaginary eigenvalues of $A$. For $n \geq 3$, the set of refined inertias$\mathbb{H}_n=\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\}$ is importantfor the onset of Hopf bifurcation in dynamical systems. We say that an $n\timesn$ sign pattern ${\cal A}$ requires $\mathbb{H}_n$ if$\mathbb{H}_n=\{\text{ri}(B) | B \in Q({\cal A})\}$. Bodine et al. conjecturedthat no $n\times n$ irreducible sign pattern that requires $\mathbb{H}_n$exists for $n$ sufficiently large, possibly $n\ge 8$. However, for each $n \geq4$, we identify three $n\times n$ irreducible sign patterns that require$\mathbb{H}_n$, which resolves this conjecture.

Sign patterns that require $\mathbb{H}_n$ exist for each $n\geq 4$