The norming graph characterization by stable involutions

Let $H$ be a bipartite graph, let $t_H(f)$ denote its homomorphism density functional, and let the stable involution group be the corresponding group of stable involutions of $H$. A graph is norming when this functional defines a norm. Norming graph conjecture. $H$ is norming if and only if it is edge-transitive under its stable involution group. The source describes this as a suspected necessary-and-sufficient characterization, analogous to the weakly norming conjectures, and gives only partial results toward it.