Herzog–Hibi conjecture on depth functions of powers of homogeneous ideals

Let $S$ be a standard graded algebra over a field $k$. For a homogeneous ideal $Q\subseteq S$, its depth function is the function $n\mapsto\operatorname{depth} S/Q^n$ for $n\geq 1$. Herzog–Hibi conjecture. Let $f:\mathbb{N}\rightarrow\mathbb{Z}_{\geq 0}$ be any function such that $f(n)=f(n+1)$ for all $n\gg 0$. Then there exists a homogeneous ideal $Q$ in a polynomial ring $S$ such that $f$ is the depth function of $Q$. The conjecture is settled affirmatively in the source paper, which constructs such an ideal for every eventually constant function $f$; thus the claim is no longer open.