Kuhlemann–Vassilevski conjecture on Laplacian eigenvalue approximation under graph disaggregation

Let $\mathsf{G}$ be a graph and let $\mathsf{G}_D$ be the graph obtained by disaggregating a vertex of $\mathsf{G}$, with weights on the internal edges of the disaggregation. Kuhlemann–Vassilevski conjecture. Under certain conditions, the Laplacian eigenvalues of $\mathsf{G}_D$ approximate the Laplacian eigenvalues of $\mathsf{G}$ when the weight on the internal edges of the disaggregation is chosen sufficiently large. The conjecture concerns when graph disaggregation preserves the spectral information of the original graph; the source does not specify the conditions or establish whether the claim has been resolved.