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We prove that generalized Bose--Einstein condensation (BEC) occurs almost surely if the intensity $\\nu_N$ of the Poisson potential satisfies $[\\ln (N)]^4/N^{1 - 2\\eta} \\ll \\nu_N \\lesssim 1$ for arbitrary $0 < \\eta \\leq 1/3$. We also show that the contact interaction alters the type of condensation, going from a type-I BEC to a type-III BEC as the strength of this interaction is increased. 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