Minimality conjecture for Hilbert depth differences of squarefree monomial ideals

Let $K$ be a field, let $S=K[x_1,\ldots,x_n]$, and let $I\subset S$ be a squarefree monomial ideal. Write

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

For each integer $d\geq 0$, let $n(d)$ and $m(d)$ be the smallest integers for which

$$\operatorname{hdepth}(S/I_{n(d),m(d)})-\operatorname{hdepth}(I_{n(d),m(d)})=d,$$

and set

$$q(d)=\operatorname{hdepth}(S/I_{n(d),m(d)}).$$

Minimality conjecture. If $q<q(d)$ or $n<n(d)$, then

$$\operatorname{hdepth}(S/I)\geq \operatorname{hdepth}(I)-d+1.$$

The conjecture asserts that the ideals $I_{n,m}$ provide the smallest examples, with respect to both the number of variables and $\operatorname{hdepth}(S/I)$, for a prescribed difference $d=\operatorname{hdepth}(S/I)-\operatorname{hdepth}(I)$. The statement remains open in the supplied source; its status is based on the authors' formulation as a proposed conjecture rather than on a resolution result.