Conjecture on bounded average degeneracy for dense random traffic-assignment instances

Let $G\sim \operatorname{RRG}(N,d)$ be a random regular graph with $N$ nodes and degree $d$, and let $D$ be a demand matrix obtained by sampling $M$ origin–destination pairs uniformly at random. Suppose that $\phi$ is convex, let $K(G,D)$ be the average degeneracy in the traffic-assignment instance $(G,D)$, and define

$$\kappa(\eta)\coloneqq \lim_{\substack{M,N\to\infty\\ M/(N(N-1))=\eta}}\operatorname{\mathbb{E}}_{G,D}K(G,D).$$

Bounded-degeneracy conjecture. The limit

$$\lim_{\eta\to\infty}\kappa(\eta)$$

exists and is finite.

This conjecture formalizes the observed plateau of path degeneracy as the demand matrix becomes dense. If true, it would help explain why the relaxed traffic-assignment problem approaches the integer problem when the number of uniformly sampled origin–destination pairs is large; the supplied text gives empirical motivation but no resolution.