Zero products of Toeplitz operators on Reinhardt domains

Let $\Omega$ be a bounded Reinhardt domain in $\mathbb{C}^n$ and $\phi_1,\ldots,\phi_m$ finite sums of quasi-homogeneous functions. We show that if the product Toeplitz operators $T_{\phi_m}\cdots T_{\phi_1}=0$ on Bergman space $\Omega$, then $\phi_j=0$ for some $j$.