Depth of some special monomial ideals

Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field. Suppose that $I$ is generated by one squarefree monomial of degree $d>0$, and other squarefree monomials of degrees $\geq d+1$. If the Stanley depth of $I/J$ is $\leq d+1$ then almost always the usual depth of $I/J$ is $\leq d+1$ too.

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