Herzog's convergence conjecture for Stanley depths of powers

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $I$ be a monomial ideal of $S$. Denote the Stanley depth of a module by ${\rm sdepth}$. Herzog's convergence conjecture. The sequence

$$\{{\rm sdepth}(I^k)\}_{k=1}^{\infty}$$

is convergent. The source says this is widely open, although it is known for complete intersections and normally torsionfree squarefree monomial ideals.