A problem on products of Toeplitz operators

A natural and interesting problem on classical Hardy space of one complex variable is the following:

Problem: If $T_{\varphi_1}T_{\varphi_2}\dotsb T_{\varphi_n}=0$, then there exist some $i$ such that $\varphi_i=0$.

In this note, we establish the kernel inclusion theorem for the products of Toeplitz operators. Using this fact, in case $n=5$, we give the above question an affirmative answer.