Let $A$ and $B$ be two full column-rank real matrices of dimension $n\times m$, $n\ge m$. Let $P$ be an $m\times m$ positive definite matrix.


> **Question.** Does there always exist a symmetric $m\times m$ matrix $X$ such that
$$
\mathrm{tr}(P(A^\top XB+B^\top X A))\ne 0\ \ ?
$$