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

Let $I$ be a monomial ideal of $S=K[x_1,\dots ,x_n]$. We show that the following are equivalent: (i) $I$ is principal, (ii) $\mathrm{hdepth}(I)=n$, (iii) $\mathrm{hdepth}(S\mathrm{/}I)=n-1$. Assuming that $I$ is squarefree, we prove that if $\mathrm{hdepth}(S\mathrm{/}I)\le 3$ or $n\le 5$ then $\mathrm{hdepth}(I)\ge \mathrm{hdepth}(S\mathrm{/}I)+1$. Also, we prove that if $\mathrm{hdepth}(S\mathrm{/}I)\le 5$ or $n\le 7$ then then $\mathrm{hdepth}(I)\ge \mathrm{hdepth}(S\mathrm{/}I)$.