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Do you have a reference for the following fact? And before that, is it true?

The Joyal model structure on simplicial sets "lifts" to a model structure on the category of marked simplicial sets, having as fibrant objects precisely the objects sent to fibrant objects by the obvious forgetful functor $\mathbf{sSet}^+\to\mathbf{sSet}$.

If it is not true, what should be a best approximation to it and where can I find it?

Thanks!

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