Asymptotic associated-prime conjecture for homological shift ideals of edge ideals

Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. For each $i\geq 0$, let $\operatorname{HS}_i(I(G)^k)$ be the $i$th homological shift ideal of the $k$th power of $I(G)$, and let $\operatorname{Ass}$ denote the set of associated prime ideals.

Asymptotic associated-prime conjecture. For all $i\geq 0$,

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

The assertion is known for $i=0$ and is proved in the paper for $i=1$; computational verification is reported for all graphs with at most seven vertices. It remains open for general $i$ and graphs.