Bilu–Linial conjecture for arbitrary graphs

Let $G$ be a graph with maximum degree $\Delta(G)>1$. An edge-signing of $G$ is a map from $E(G)$ to $\{-1,1\}$; write $G^\sigma$ for the resulting edge-signed graph and let $\rho(G^\sigma)$ be the maximum absolute value of an eigenvalue of its signed adjacency matrix. Bilu–Linial conjecture for arbitrary graphs. There exists an edge-signing $\sigma$ of $G$ such that

$$\rho(G^\sigma)\leq 2\sqrt{\Delta(G)-1}.$$

This is presented as a stronger conjecture obtained by omitting regularity from the regular-graph formulation, and is attributed to reference greg. The source gives no resolution status.