The p-modular Hadamard conjecture

Let $p$ be an odd prime, and let a $p$-modular Hadamard matrix of order $d$ mean the matrix notion used in the paper. The $p$-modular Hadamard conjecture. For all but finitely many positive integers $d$, a $p$-modular Hadamard matrix of order $d$ exists if and only if $d$ is even or $d$ is a quadratic residue modulo $p$. This conjecture asserts asymptotic sufficiency of the stated necessary condition and is used to motivate further finite-field SIC constructions; the paper does not establish it in general.