An upper bound on stability of powers of matroidal ideals

Let $R=K[x_1,\dots ,x_n]$ be a polynomial ring in $n$ variables over a field $K$ and $I$ be a matroidal ideal of degree $d$. Let $\text{astab}(I)$ and $\text{dstab}(I)$ be the smallest integers $l$ and $k$, for which $\text{Ass}(I^l)$ and $\text{depth}(R\mathrm{/}{I}^k)$ stabilize, respectively. In this paper, we show that $\text{astab}(I),\text{dstab}(I)\le \min \{d,\mathrm{\ell }(I)\}$, where $\mathrm{\ell }(I)$ is the analytic spread of $I$. Furthermore, by a counterexample we give a negative answer to the conjecture of Herzog and Qureshi \cite{HQ} about stability of matroidal ideals.