The associated-prime persistence conjecture for homological shift ideals

Let $S$ be the polynomial ring in the paper, let $I\subset S$ be a polymatroidal ideal, and let $\operatorname{HS}_i(I^k)$ denote the $i$th homological shift ideal of $I^k$. Associated-prime persistence conjecture. For every $i>0$, the associated-prime sequence is an increasing chain beginning at the $i$th power:

$$\operatorname{Ass}\,\operatorname{HS}_i(I^{i})\subset\operatorname{Ass}\,\operatorname{HS}_i(I^{i+1})\subset\operatorname{Ass}\,\operatorname{HS}_i(I^{i+2})\subset\cdots.$$

The assertion is proved in the paper for principal Borel ideals, polymatroidal ideals satisfying the strong exchange property, and matroidal edge ideals; it remains open for arbitrary polymatroidal ideals.