Universal spectral threshold dominance

Prove or disprove that every graph is spectrally threshold dominated: for every graph G and every integer k, determine whether there exists a threshold graph of the same order and size whose sum of the k largest Laplacian eigenvalues is at least the corresponding sum for G.

Background

A graph is spectrally threshold dominated if, for every k, a threshold graph with the same order and size majorizes the top-k Laplacian eigenvalue sum of the graph. This property is known for split graphs and is preserved under certain disjoint-union and complement operations.

The conjecture would imply that threshold graphs maximize Laplacian energy among all graphs with fixed order and size, and would connect universal spectral threshold dominance with Brouwer-type Laplacian eigenvalue-sum inequalities. The paper states that the conjecture remains unresolved in general.