C^⁎-embedding and P-embedding in subspaces of products of ordinals

It is known that in X=A×B, where A and B are subspaces of ordinals, all closed C^⁎-embedded subspaces of X are P-embedded. Also it is asked whether all closed C^⁎-embedded subspaces of X are P-embedded whenever X is a subspace of products of two ordinals. In this paper, we prove that both of the following are consistent with ZFC: • there is a subspace X of (ω+1)×ω₁ such that the closed subspace X∩({ω}×ω₁) is C^⁎-embedded in X but not P-embedded in X, • for every subspace X of (ω+1)×ω₁, if the closed subspace X∩({ω}×ω₁) is C^⁎-embedded in X, then it is P-embedded in X.