On the DJL conjecture for order 6

Abstract

In 1994 Drew, Johnson and Loewy conjectured that for $n \ge 4$, the cp-rank of any $n\times n$ completely positive matrices is at most $\lfloor{n^2}/{4}\rfloor$. Recently this conjecture has been proved for $n=5$ and disproved for $n\ge 7$, leaving the case $n=6$ open. We make a step toward proving the conjecture for $n=6$. We show that if $A$ is a $6\times 6$ completely positive matrix that is orthogonal to an exceptional extremal copositive matrix, then the cp-rank of $A$ is at most $9$.