Hilbert-depth comparison for squarefree monomial ideals in at most nine variables

Let $S=K[x_1,\ldots,x_n]$ and let $I\subset S$ be a squarefree monomial ideal. The nine-variable Hilbert-depth conjecture. If

$$n\leq 9,$$

then

$$\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I).$$

The paper gives examples showing failure of the comparison in larger polynomial rings, while its results establish the inequality in several smaller cases. Whether it holds for every squarefree monomial ideal with at most nine variables remains open.