The multi-edge Laplacian matching root integral variation conjecture

Let $G$ be a connected graph, let $F\subseteq E(G^c)$ be nonempty, and set $H=G+F$. Let $\lambda_1(G),\dots,\lambda_n(G)$ denote the roots of the Laplacian matching polynomial of $G$. Multi-edge LMRIV conjecture. Multi-edge Laplacian matching root integral variation does not occur for $G$: there are no integers $t_1,\dots,t_n\in\mathbb Z_{\geq 0}$ with $t_1+\cdots+t_n=2|F|$ such that, as multisets,

$$\{\lambda_1(H),\dots,\lambda_n(H)\}=\{\lambda_1(G)+t_1,\dots,\lambda_n(G)+t_n\}.$$

The conjecture has been verified by computer for $n\leq 9$; it generalizes the one-edge patterns corresponding to one-place and two-place LMRIV and remains open in general.