Stanley depth of the integral closure of monomial ideals

Let $I$ be a monomial ideal in the polynomial ring $S=\mathbb{K}[x_1,...,x_n]$. We study the Stanley depth of the integral closure $\bar{I}$ of $I$. We prove that for every integer $k\geq 1$, the inequalities ${\rm sdepth} (S/\bar{I^k}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (\bar{I^k}) \leq {\rm sdepth} (\bar{I})$ hold. We also prove that for every monomial ideal $I\subset S$ there exist integers $k_1,k_2\geq 1$, such that for every $s\geq 1$, the inequalities ${\rm sdepth} (S/I^{sk_1}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (I^{sk_2}) \leq {\rm sdepth} (\bar{I})$ hold. In particular, $\min_k \{{\rm sdepth} (S/I^k)\} \leq {\rm sdepth} (S/\bar{I})$ and $\min_k \{{\rm sdepth} (I^k)\} \leq {\rm sdepth} (\bar{I})$. We conjecture that for every integrally closed monomial ideal $I$, the inequalities ${\rm sdepth}(S/I)\geq n-\ell(I)$ and ${\rm sdepth} (I)\geq n-\ell(I)+1$ hold, where $\ell(I)$ is the analytic spread of $I$. Assuming the conjecture is true, it follows together with the Burch's inequality that Stanley's conjecture holds for $I^k$ and $S/I^k$ for $k\gg 0$, provided that $I$ is a normal ideal.