Sign patterns that require $mathbb{H}_n$ exist for each $ngeq 4$

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 of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pure imaginary 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 important for the onset of Hopf bifurcation in dynamical systems. We say that an $ntimes n$ sign pattern ${cal A}$ requires $mathbb{H}_n$ if $mathbb{H}_n={text{ri}(B) | B in Q({cal A})}$. Bodine et al. conjectured that no $ntimes n$ irreducible sign pattern that requires $mathbb{H}_n$ exists for $n$ sufficiently large, possibly $nge 8$. However, for each $n geq 4$, we identify three $ntimes n$ irreducible sign patterns that require $mathbb{H}_n$, which resolves this conjecture.