Abrishami's common Laplacian eigenvalues conjecture for equivalent cographs

Let $G$ and $H$ be equivalent cographs with reductions $R_G$ and $R_H$. Suppose the vertices of $R_G$ and $R_H$ are labeled by the twin numbers $t_i$ of the $k$ twin classes they represent. Here, a twin class is a class of vertices represented by one vertex in the reduction, and its type records the corresponding type in the cograph.

Abrishami's conjecture. $G$ and $H$ have at least

$$k + \sum_{i\in I}(t_i-1)$$

Laplacian eigenvalues in common, where $I$ is the set of indices of the twin classes whose types are identical in $G$ and $H$.

This conjecture concerns how much of the Laplacian spectrum is preserved between equivalent cographs. The supplied source attributes it to Tara Abrishami's master's thesis, but gives no resolution, so its status remains open.