Characterization of the edge-universality regime for dense random regular graphs

Let $d$ and $N$ be parameters of the random regular graph model, with $d$ sufficiently large and $N\geqslant 2d$. Let (1.2) denote the edge-universality assertion referred to in the source.

Edge-universality regime conjecture. The assertion in (1.2) holds if and only if

$$N\ll d^3.$$

This is presented as a further consequence conditional on combining the methods of the cited work with the current paper; its precise interpretation depends on the preceding definition of (1.2).