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

$\underset{n\to \mathrm{\infty }}{\lim }{\text{Prob}}_{\sigma \in {{𝒮}}_n^{E_B}}[\text{NewSpec}(B[\sigma ])\subset [-\rho -\epsilon ,\rho +\epsilon ]]=1$

for any ε > 0.