The duplication lower-bound conjecture for Betti numbers of symbolic powers

Let $G$ be a graph on $m$ vertices and let $\alpha=(\alpha_1,\dots,\alpha_m)\in(\mathbb Z^+)^m$, where $G^\alpha$ is obtained by duplicating the vertices of $G$ according to $\alpha$. Write $I(G)$ for the edge ideal of $G$, $I(G)^{(s)}$ for its $s$-th symbolic power, and $\beta_{i,j}$ for the graded Betti numbers. Duplication lower-bound conjecture. For all $i,j\in\mathbb Z^+$ and $s\geq 2$,

$$\beta_{i,j}(I(G^\alpha)^{(s)})\geq\left(\prod_i\alpha_i\right)\beta_{i,j}(I(G)^{(s)}).$$

The conjecture proposes that syzygies of symbolic powers persist under vertex duplication in at least $\prod_i\alpha_i$ disjoint substitution patterns; the paper notes that this stronger bound should hold over any field, but does not establish it.