Conjecture on truncated generalized block Laplacian spectra

Let $G$ be a graph on $n$ vertices, with distinct degrees $\delta_1 > \delta_2 > \cdots > \delta_p$. Set $r = \lceil\log_2 n\rceil + 1$, partition $V = \bigsqcup_{i=1}^r V_i$ by taking $V_i$ to be the vertices of degree $\delta_i$ for $i=1,\ldots,r-1$ and $V_r = V\setminus\bigsqcup_{i=1}^{r-1}V_i$, and let $e_i$ be the characteristic vector of $V_i$ and $J_{i,j}=e_i e_j^\top$. Define the truncated generalized block Laplacian spectrum using $\bm{A}=(A_G,J_{1,1},J_{1,2},\ldots,J_{r,r})$. The truncated generalized block Laplacian conjecture. Almost all graphs are determined by their truncated generalized block Laplacian spectra. The claim proposes that this multivariate spectral invariant distinguishes almost every graph up to isomorphism; the source presents it as a conjecture for future research and gives only experimental evidence, so its resolution remains open.