Let $S=k[x_1,\dots,x_n]$ and $J\subset I$ two squarefree monomial ideals in $S$ such that $I$ is generated in degree $d$, and $J$ is generated in degree $d+1$. Let $I=(f_1,\ldots,f_r),\quad I'=(f_1,\ldots,f_e),e\leq r$ and $J'=I'∩J$. If we have $\operatorname{depth}(I'/J')=d+1$, and$\operatorname{depth}(S/I)=\operatorname{depth}(S/J)=\operatorname{depth}(S/I')=\operatorname{depth}(S/J')=\operatorname{depth}(S/(I'+J))=d,$ is it possible to have $\operatorname{depth}(I/J)>d+1$ ?
I think that it isn't true, but I don't see why. (Here $\operatorname{depth}(I/J)$ is defined from Depth Lemma.)