The two-place Laplacian matching root integral variation conjecture

Let $G$ be a connected graph and let $e$ be a non-edge of $G$. Write the nonnegative real roots of the Laplacian matching polynomial $4\mathscr{LM}_G(x)\4$ in non-increasing order as $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$. Two-place LMRIV conjecture. When $e$ is added to $G$, two-place Laplacian matching root integral variation does not occur; that is, it is not the case that exactly two roots increase by $1$ while all remaining roots stay unchanged. This extends the known impossibility result for graphs satisfying $g(G)/c(G)>7/6$, where $g(G)$ is the girth and $c(G)$ is the dimension of the cycle space, and remains open for general connected graphs.