Comparing Hilbert depth of $I$ with Hilbert depth of $S/I$. III

Let $I$ be a squarefree monomial ideal of $S=K[x_1,\ldots,x_n]$.

We prove that if $\operatorname{hdepth}(S/I)\leq 8$ or $n\leq 10$ then $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)-1$.