Seyed Fakhari's Stanley depth conjecture for integrally closed monomial ideals

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $I\subset S$ be an integrally closed monomial ideal. If $\ell(I)$ denotes the analytic spread of $I$, then Seyed Fakhari's conjecture.

$$\operatorname{sdepth}(S/I)\geq n-\ell(I)\qquad\text{and}\qquad \operatorname{sdepth}(I)\geq n-\ell(I)+1.$$

These inequalities are motivated by Burch's inequality for the depth of powers of ideals and by the relationship between analytic spread and asymptotic depth. The paper gives examples showing that they fail for arbitrary monomial ideals, but reports no counterexample among integrally closed monomial ideals; the conjecture is also used to obtain Stanley's conjecture asymptotically for powers of normal monomial ideals.