The Bapat-Sunder eigenvalue conjecture fails over the reals

In 1986 Bapat and Sunder asked whether per(??) is the largest eigenvalue of F?? = [????,?? per(??(??, ??))] for every positive semidefinite Hermitian ??, where ??(??, ??) deletes row ?? and column ??. Drury disproved this over the complex field in 2018 and conjectured that the property holds for real positive semidefinite matrices. We disprove Drury's conjecture with a rank-4, 32 × 32 integer Gram matrix and an integer vector whose Rayleigh quotient exceeds per(??); verification uses integer arithmetic. Together, the two counterexamples resolve the Bapat–Sunder eigenvalue conjecture negatively over both fields. Counterexamples exist of every order ?? ≥ 32. As an application to quantum optics, the violation yields a 36-mode orthogonal interferometer and a direction of relative arrival-time delays among 32 photons with identical-variance Gaussian wave packets along which every sufficiently small nonzero delay increases the probability that all photons are detected within four designated output modes.