Homological shift generation conjecture for complementary edge ideals

Let $G$ be a connected graph, and let $I=I_c(G)$ be its complementary edge ideal. For each $i\geq 0$, let $\HS_i(\mathcal{R}(I))$ denote the $i$-th homological shift ideal of the Rees algebra of $I$. Homological shift generation conjecture. The ideal $\HS_i(\mathcal{R}(I))$ is generated in degree at most $i$.

This conjecture predicts a uniform degree bound for the homological shifts of Rees algebras arising from complementary edge ideals. The source states that it is proved for the relevant graph classes considered in the paper; it also records that, for polymatroidal ideals, the case $i=1$ is known from the cited work of Fakhari and Lemmens.