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We prove that for almost every choice of $(\\lambda_1, \\ldots, \\lambda_k)$ in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). 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We prove that for almost every choice of $(\\lambda_1, \\ldots, \\lambda_k)$ in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). 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