The edge-transitivity characterization of weakly norming graphs

Let $H$ be a bipartite graph. Its cut involution group is the group of graph symmetries generated by cut involutions, and $H$ is weakly norming when its associated homomorphism-density functional defines a norm. Weakly norming graph conjecture. $H$ is weakly norming if it is edge-transitive under its cut involution group. The paper presents this as an open characterization problem; the condition is sufficient in the authors' examples, but its necessity and sufficiency were not established.