Eventual constancy conjecture for Serre-depth functions of powers

Let $(R,\mathfrak{m})$ be a Noetherian local ring or a standard graded $K$-algebra. Let $I\subset R$ be an ideal, homogeneous when $R$ is a $K$-algebra. The Serre-depth stabilization conjecture. asserts that

$$S_r\text{-}\!\operatorname{depth}(R/I^{k+1})=S_r\text{-}\!\operatorname{depth}(R/I^k),$$

for all $r\ge1$ and all $k\gg0$. This extends Brodmann's eventual constancy theorem for ordinary depth; the paper proves the assertion for monomial ideals in a standard graded polynomial ring, while the stated general form remains open.