Herzog–Qureshi's equality conjecture for polymatroidal ideals

Let $I$ be a polymatroidal ideal. Write $\astab(I)$ for the index of stability of the associated-prime sets of the powers of $I$, and $\dstab(I)$ for the index of stability of their depth functions.

Herzog–Qureshi's conjecture. Does

$$\astab(I)=\dstab(I)$$

hold for every polymatroidal ideal $I$?

This asks whether the stabilization index of the associated primes always equals the stabilization index of the depth function for powers of a polymatroidal ideal. The supplied context attributes the question to Herzog and Qureshi but gives no evidence of a resolution, so its status is open.