From seven to eleven: Completely positive matrices with high cp-rank

We study n×nn×n completely positive matrices MM on the boundary of the completely positive cone, namely those orthogonal to a copositive matrix SS which generates a quadratic form with finitely many zeroes in the standard simplex. Constructing particular instances of SS, we are able to construct counterexamples to the famous Drew–Johnson–Loewy conjecture (1994) for matrices of order seven through eleven.