The answer for a general $n$ is positive: the discriminant is a sum of squares of polynomials in the entries of $H$. The first formula was given by Ilyushechkin and involves $n!$ squares. This number was improved by Domokos into $$\binom{2n-1}{n-1}-\binom{2n-3}{n-1}.$$ See Exercise #113 on my page.
Denis Serre
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