One more counterexample on sign patterns

Abstract

The \textit{sepr-sequence} of an $n\times n$ real matrix $A$ is $(s_1,\ldots,s_n)$, where $s_k$ is the subset of those signs of $+,-,0$ that appear in the values of the $k\times k$ principal minors of $A$. The $12\times 12$ matrix $$\left(\begin{array}{cccccc|ccc|ccc} 0&0&0&0&0&0&0&0&0&a_1&0&0\\ 0&0&0&0&0&0&0&0&0&0&a_2&0\\ 0&0&0&0&0&0&0&0&0&0&0&a_3\\ 0&0&0&0&0&0&0&0&0&0&0&a_4\\ 0&0&0&0&0&0&0&0&0&0&0&a_5\\ 0&0&0&0&0&0&0&0&0&0&0&a_6\\\hline b_1&b_2&0&0&0&0&0&0&0&0&0&0\\ b_3&b_4&0&0&b_5&-b_6&0&0&0&0&0&0\\ 0&b_7&b_8&-b_9&b_{10}&b_{11}&0&0&0&0&0&0\\\hline 0&0&0&0&0&0&c_1&0&0&0&0&0\\ 0&0&0&0&0&0&0&c_2&0&0&0&0\\ 0&0&0&0&0&0&0&0&c_3&0&0&0 \end{array}\right)$$ does always have $s_k=\{0,+,-\}$ if $k=3,6,9$ and $s_k=\{0\}$ otherwise, provided that the variables are positive. However, every principal $9\times 9$ minor that is not identically zero can take values of both signs.