Bi-continuity of the grapheur–graphon duality pairing

Prove that the duality pairing between symmetric grapheurs and graphons is bi-continuous: whenever a sequence of symmetric grapheurs converges to a grapheur and a sequence of graphons converges to a graphon, the corresponding random pairing variables converge weakly.

Background

The paper defines a random pairing between symmetric grapheurs and graphons and proves separate continuity characterizations: convergence of grapheurs can be detected by pairing them with step graphons, while convergence of graphons can be detected by pairing them with grapheurs associated with finite undirected graphs. The authors conjecture the stronger joint continuity property in which both arguments vary simultaneously.

If true, the conjecture would yield characterizations of convergence in either space using pairings against every fixed element of the dual space, strengthening the duality theorem established in the paper.