### ${\mathcal{D}}_{n,r}$ is not potentially nilpotent for $n\ge 4r-2$

An $n\times n$ sign pattern $\mathcal{A}$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $\mathcal{A}$. Let ${\mathcal{D}}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$$(i=1...$