Zero products of Toeplitz operators on Reinhardt domains

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