Herzog's polarization conjecture for Stanley depth

Let $R=\mathbb{K}[X_1,\ldots,X_n]$ be a polynomial ring, let $I\subset R$ be a monomial ideal, and let $I^p\subset R^p$ be its polarization, where $R^p$ is the polynomial ring in which $I^p$ is defined. Herzog's polarization conjecture.

$$\operatorname{sdepth} R/I-\operatorname{depth} R/I=\operatorname{sdepth} R^p/I^p-\operatorname{depth} R^p/I^p.$$

The conjecture predicts that polarization preserves the difference between Stanley depth and ordinary depth. It is stated here as a conjecture by Herzog, but the paper's abstract says that the authors prove it; the supplied span itself does not record that resolution.