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We show that the monodromy group of a good family for $\\mathcal{M}_{n,2n+2}$ is Zariski dense in the corresponding symplectic or orthogonal group if $n\\geq 3$. In particular, the period map does not give a uniformization of any partial compactification of the coarse moduli space as a Shimura variety whenever $n\\geq 3$. This disproves a conjecture of Dolgachev. 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