Sandbox:DavidKohler/Relativized Alon Conjecture/Strong conjecture

This strong version of the conjecture has been first suggested by Friedman in ?? (to check)

Conjecture

Let B be a graph and let ρ be the spectral radius of the universal cover of the the graph B, then

$\lim _{n\to \infty }{\text{Prob}}_{\sigma \in {\mathcal {S}}_{n}^{E_{B}}}{\big [}{\text{NewSpec}}(B[\sigma ])\subset [-\rho -\varepsilon ,\rho +\varepsilon ]{\big ]}=1$

for any ε > 0.