The asymptotic two-value conjecture for the Hilbert depth of $I_{n,n}$

Let $S$ be the polynomial ring and $I_{n,n}$ the special class of squarefree monomial ideals considered in the paper, and let $\operatorname{hdepth}(S/I_{n,n})$ denote the Hilbert depth of its quotient. Based on computer experiments, Hilbert-depth two-value conjecture. There exists a constant $\alpha\approx 1.817$ such that

$$\operatorname{hdepth}(S/I_{n,n})\in \{\left\lfloor \alpha n \right\rfloor - 2,\left\lfloor \alpha n \right\rfloor - 1\},\qquad n\geq 2.$$

The claim predicts that the Hilbert depth is confined to two consecutive values determined by a linear function of $n$. The parser supplies no evidence that the conjecture has been resolved, so its status is left open.