Chung's singular-value discrepancy conjecture for regular graphs

For a graph $G$ on $n$ vertices, write $e(G)$ for its number of edges and set

$$\rho(G)=e(G)\binom{n}{2}^{-1}.$$

For nonempty vertex subsets $X,Y\subseteq V(G)$, let

$$disc_{2}(G)=\max_{X,Y\subseteq V(G),\,X\neq\varnothing,\,Y\neq\varnothing}\frac{\left|e(X,Y)-\rho(G)|X||Y|\right|}{\sqrt{|X||Y|}}.$$

Let $\sigma_{2}(G)$ denote the second-largest singular value of the adjacency matrix of $G$. Chung's conjecture. There is an absolute constant $C$ such that for every regular graph $G$,

$$\sigma_{2}(G)<C\,disc_{2}(G).$$

The conjecture asks for a direct comparison between spectral expansion and edge-distribution discrepancy in regular graphs. The source attributes it to Fan Chung; its resolution is not specified in the supplied text.