Expressiveness hierarchy conjecture for Local k-FGNN

For an integer $k\ge 2$, let Local $k$-FGNN, Local $k$-GNN, and $k$-FGNN denote the corresponding higher-order graph neural network architectures, ordered by their ability to distinguish graphs. Local $k$-FGNN expressiveness conjecture. For all $k\ge 2$, Local $k$-FGNN is strictly more expressive than Local $k$-GNN and strictly less expressive than $k$-FGNN. Establishing this would complete the paper's expressiveness hierarchy for the higher-order local architectures and provide a simple characterization of the currently unresolved Local $k$-FGNN case. A similar open question was raised informally by Zhang et al. (2023).