Abstract
The Drury permanent conjecture, published by Zhang in 2016 and disproved over the complex numbers by Hutchinson in 2017, has remained open in the real case, where a proof would imply the real case of the Chollet conjecture. The real 5 Γ 5 counterexample reported by Paksoy in 2023 concerns a different inequality, whose principal submatrices have order π β 1 rather than order π β 2 as in the conjecture; it is verified that Paksoy's matrix satisfies the conjectured inequality strictly, under every simultaneous permutation of its rows and columns. The real case is then settled in the negative: an 8 Γ 8 real positive semidefinite matrix with integer entries and rank 3 is exhibited that violates the inequality, with an exact integer certificate of the violation. Direct sums with identity matrices give counterexamples of every order π β₯ 8.
Article accepted for publication in Electronic Journal of Linear Algebra (International Linear Algebra Society).