Incomparable copies of a poset in the Boolean lattice

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Let Bₙ be the poset generated by the subsets of [n] with the inclusion as relation and let P be a finite poset. We want to embed P into Bₙ as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets P as n n/2/M(P), where M(P) denotes the minimal size of the convex hull of a copy of P. We discuss both weak and strong (induced) embeddings.