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Let $sSet^2$ be the category of bisimplicial sets.

In the diagonal model structure on $sSet^2$ weak equivalences are diagonal weak equivalence (i.e.$ X \rightarrow Y$ is a weak equivalence if $dX \rightarrow dY$ is a weak equivalence of simplicial sets) and cofibrations are monomorphisms. Let us call the fibrations in this model structure as diagonal fibrations.

In Moerdijk model structure on $sSet^2$, $X \rightarrow Y$ is a fibration (weak equivalence) if $dX \rightarrow dY$ is a kan fibration (weak equivalence of simplicial sets).

Is there any condition known under which a Moerdijk fibrant $X$ is diagonal fibrant?

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