Hermitian rank in ideal powers

We prove that the (hermitian) rank of $QP^d$ is bounded from below by the rank of $P^d$ whenever $Q$ is not identically zero and real-analytic in a neighborhood of some point on the zero set of $P$ in $\mathbb{C}^n$ and $P$ is a polynomial of bidegree at most $(1,1)$. This result generalizes the theorem of D'Angelo and the second author which assumed that $P$ was bihomogeneous. Examples show that no hypothesis can be dropped.