On stability properties of powers of polymatroidal ideals

Let $R=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ be the polynomial ring in $n$ variables over a field $K$ with the maximal ideal $\mathfrak{m}=(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$. Let $\text{astab}(I)$ and $\text{dstab}(I)$ be the smallest integer $n$ for which $\text{Ass}(I^n)$ and $\text{depth}(I^n)$ stabilize, respectively. In this paper we show that $\text{astab}(I)=\text{dstab}(I)$ in the following cases: \begin{itemize} \item[(i)] $I$ is a matroidal ideal and $n\le 5$. \item[(ii)] $I$ is a polymatroidal ideal, $n=4$ and $\mathfrak{m}\mathrm{\notin }{\text{Ass}}^{\mathrm{\infty }}(I)$, where ${\text{Ass}}^{\mathrm{\infty }}(I)$ is the stable set of associated prime ideals of $I$. \item[(iii)] $I$ is a polymatroidal ideal of degree $2$. \end{itemize} Moreover, we give an example of a polymatroidal ideal for which $\text{astab}(I)\mathrm{\ne }\text{dstab}(I)$. This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.