Let $A$ be an $n \times n$. Define $A_{-i}$ to be the matrix $A$ without the $i$-th column and row. For instance $$ A= \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix} \implies A_{-2}= \begin{pmatrix} 1 & 3\\ 7 & 9 \end{pmatrix} $$
Is it possible to find $S$ symmetric and not positive semidefinite such that there exist $i_1\not= i_2$ with $A_{-i_1}, A_{-i_2}$ which are positive semidefinite?
$$ \left( \begin{array}{ccc} 5 & -3 & -3 \\ -3 & 5 & -3 \\ -3 & -3 & 5 \\ \end{array} \right) $$
$$ \left( \begin{array}{ccc} 7 & -4 & -5 \\ -4 & 8 & -6 \\ -5 & -6 & 9 \\ \end{array} \right) $$