Comparing Hilbert depth of $I$ with Hilbert depth of $S\mathrm{/}I$. III

Let $I$ be a squarefree monomial ideal of $S=K[x_1,\dots ,x_n]$. We prove that if $\mathrm{hdepth}(S\mathrm{/}I)\le 8$ or $n\le 10$ then $\mathrm{hdepth}(I)\ge \mathrm{hdepth}(S\mathrm{/}I)-1$.