Monotonicity conjecture for the Hilbert depth of lexsegment ideals

Let $S=\mathbb{K}[x_1,\dots,x_n]$, and let $I\subset S$ be a lexsegment ideal generated by monomials of degree $d$. Write $\mu(I)$ for the number of minimal generators of $I$. Lexsegment Hilbert-depth monotonicity conjecture. The function $\operatorname{hdepth}_1(I)$ is decreasing as a function of $\mu(I)$.

Such a monotonicity statement would describe how the standard graded Hilbert depth changes as the number of generators of a fixed-degree lexsegment ideal increases. The source presents it as a conjectural belief supported by computational evidence, and no resolution is supplied.