Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence. Assume $depth \frac{R}{I+(\ell_1, \ldots, \ell_{t})}=0.$ Then $depth (R/I)\le t?$

Thanks so much for any suggestions.