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Then, inspired by results of Hern\\'andez-Guti\\'errez, Hru\\v{s}\\'ak and van Mill, using a technique of Medvedev, we construct a non-Polish subspace $X$ of $2^\\omega$ such that $X^\\omega$ is countable dense homogeneous. This gives the first $\\mathsf{ZFC}$ answer to a question of Hru\\v{s}\\'ak and Zamora Avil\\'es. 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