P (R), ordered by homeomorphic embeddability, does not represent all posets of cardinality 2^c

We prove it to be consistent that there is a poset of cardinality 2^c which is not realizable in P (R), ordered by homeomorphic embeddability. This addresses and answers resolutely (and in the negative) the open question of whether there is a ZFC theorem that all posets of cardinality 2^c can be represented by subspaces of the real line ordered by homeomorphic embeddability. This question arises from the pioneering work of Banach, Kuratowski and Sierpiński in the area and this result complements the recent work of [A.E. McCluskey, D. Shakhmatov, It is consistent that all posets of cardinality 2^c can be realized within P (R), preprint], thereby providing a proof of independence.