Nakano-positive determinants outside the Hodge-Riemann cone

Dinh and Nguyên asked whether the determinant of a Griffiths positive matrix with $(1,1)$-form entries belongs to the Hodge–Riemann cone. For every $n⩾4$ and $2⩽k⩽n-2$, we present an explicit Nakano positive $k\times k$ matrix of constant $(1,1)$-forms on ${\text{C}}^n$ whose determinant has a singular Lefschetz map in bidegree $(1,n-k-1)$. This gives a negative answer throughout this range. The boundary cases $n=k⩾4$ and $n=k+1⩾5$ remain open. We next study the question under the simultaneous diagonalizability (SD) condition. Under this condition, we prove the Hodge–Riemann property in every bidegree $(p,q)$ with $p+q=n-k$ and $\min (p,q)⩽1$. Consequently, SD gives an affirmative answer when $1⩽k=n-2$ or $n-3$. For every $n⩾6$ and $2⩽k⩽n-4$, however, we present SD examples whose Lefschetz map in bidegree $(2,n-k-2)$ is singular, showing that SD alone does not imply the Hodge–Riemann property in all bidegrees. The positive result uses the theory of dually Lorentzian polynomials developed by Ross, Süß, and Wannerer, in particular their generalized Alexandrov–Fenchel inequality and its equality characterization. Exact Python verification programs accompany the constructions.