  Let G and H be equivalent cographs with their reduction R_G and R_H, and
suppose the vertices of R_G and R_H are labeled by the twin numbers t_i of the
k twin classes they represent. In this paper, we prove that G and H have at
least k + \sum_{i\in I}(t_i-1) Laplacian eigenvalues in common, where I is the
indices of the twin classes whose types are identical in G and H. This confirms
the conjecture proposed by T. Abrishami \cite{Abris}. We also show that no two
nonisomorphic equivalent cographs are L-cospectral.
