When will the Stanley depth increase

Let $I\subset S=\text{KK}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ be an ideal generated by squarefree monomials of degree $\ge d$. If the number of degree $d$ minimal generating monomials ${\mu }_d(I)\le \min (\left(\genfrac{}{}{0px}{}{n}{d+1}\right),{\sum }_{j=1}^{n-d}\left(\genfrac{}{}{0px}{}{2j-1}{j}\right))$, then the Stanley depth ${\text{sdepth}}_S(I)\ge d+1$.