Conjecture on the normalized depth function of squarefree powers

Let $I$ be a squarefree monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$. For each $k$ for which $I^{[k]}\neq 0$, let $d_k$ be the minimum degree of a monomial in the minimal generating set $G(I^{[k]})$, and define the normalized depth function by

$$g_I(k)=\operatorname{depth}(S/I^{[k]})-(d_k-1).$$

Normalized-depth conjecture. The function $g_I(k)$ is nonincreasing.

The paper presents this as a conjecture motivated by computations and examples, concerning the behavior of depths of squarefree powers; the supplied material gives no resolution, so the conjecture remains open here.