  Let $K$ be a field, $R=K[X_1, ..., X_n]$ be the polynomial ring and $J
\subsetneq I$ two monomial ideals in $R$. In this paper we show that
$\mathrm{sdepth}\ {I/J} - \mathrm{depth}\ {I/J} = \mathrm{sdepth}\
{I^p/J^p}-\mathrm{depth}\ {I^p/J^p}$, where $\mathrm{sdepth}\ I/J$ denotes the
Stanley depth and $I^p$ denotes the polarization. This solves a conjecture by
Herzog and reduces the famous Stanley conjecture (for modules of the form
$I/J$) to the squarefree case. As a consequence, the Stanley conjecture for
algebras of the form $R/I$ and the well-known combinatorial conjecture that
every Cohen-Macaulay simplicial complex is partitionable are equivalent.
