Fakhari’s conjecture on Stanley depth of integrally closed monomial ideals

Let $\mathbb{K}$ be a field, let $S=\mathbb{K}[x_1,\ldots,x_n]$, and let $I\subset S$ be an integrally closed monomial ideal. Let $\ell(I)$ denote the analytic spread of $I$. Fakhari’s conjecture.

$$\operatorname{sdepth}(S/I)\geq n-\ell(I)$$

and

$$\operatorname{sdepth}(I)\geq n-\ell(I)+1.$$

The conjecture proposes analytic-spread lower bounds for the Stanley depths of the quotient and the ideal. The paper proves these bounds in particular cases, including squarefree monomial ideals generated in a single degree, but does not state that the conjecture is resolved in full.