If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff gaps and limits". In the paper "R. Frankiewicz, Some remarks on embeddings of boolean algebras and topological spaces II, Fund. Math, 126 (1), 1985, 63-68.", the author showed that it is consistent with MA+¬CH the existence of a boolean algebra with cardinality equal to the continuum which does not embed in ℘(ω)/Fin.
Is it consistent with $MA+\neg CH$ that every boolean algebra with cardinaly equal to the continuum embeds in ℘(ω)/Fin?