We study $n\times n$ completely positive matrices $M$ on the boundary of the completely positive cone, namely those orthogonal to a copositive matrix $S$ which generates a quadratic form with finitely many zeroes in the standard simplex. Constructing particular instances of $S$, we are able to construct counterexamples to the famous Drew-Johnson-Loewy conjecture (1994) for matrices of order seven through eleven.
Preprint NI14007-POP, Isaac Newton Institute for Mathematical Sciences, University of Cambridge UK, accepted for publication in: Linear Alg. Appl.
View From seven to eleven: completely positive matrices with high cp-rank