A $\sqrt{2}$-Approximation to the Bilu-Linial Conjecture

Bilu and Linial conjectured that if $d\ge 2$, then every $d$-regular graph $G$ has an edge signing $σ:E(G)\to\{-1,1\}$ such that its signed adjacency matrix $A_σ$ satisfies the Ramanujan bound \begin{equation*} ρ(A_σ)\le 2\sqrt{d-1} \end{equation*} and they proved that \begin{equation*} ρ(A_σ)=O