Efficient-ordering conjecture for powers of edge ideals

Let $G$ be a graph with edge ideal $I(G)$. Suppose that, for some integer $q\in\mathbb{N}$, $I(G)^q$ has linear quotients with an ordering $\mathcal N^{(q)}:u_1>\dots>u_r$ of its generators. Recursively construct the efficient ordering $\mathcal N^{(s)}$ from $\mathcal N^{(q)}$ as defined in the source. Efficient-ordering conjecture. Then $I(G)^s$ has linear quotients with the efficient ordering $\mathcal N^{(s)}$ for every integer $s\geq q$. This is a refinement of the persistence conjecture because it specifies the ordering intended to provide linear quotients at every higher power; it remains open.