Let $P_1,\ldots, P_d, Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots, x_n]$ be homogenous polynomials of degree at most $r$.

Assume that $P_1 \cdot P_2 \cdots P_{d-1} \cdot P_d \in \langle Q_1, \ldots, Q_k \rangle$. Here $\langle h_1, \ldots, h_s \rangle$ is the ideal with the generators $h_1, \ldots, h_s$.

Is it true that for some $\{ i_1, \ldots, i_f \} \subseteq \{1, \ldots, d\}$ the polynomial $P_{i_1} \cdots P_{i_f} \in \langle Q_1, \ldots, Q_k \rangle$, where $f= f(k,r)$?