I am looking for an example of a compact complex analytic space, reduced and irreducible, which does not admit any holomorphic embedding into any (smooth) complex manifold (possibly non-compact).

I am aware of many examples of a similar kind of algebraic varieties which cannot be embedded in any smooth scheme, see for example this question and references there (in particular, the paper of Roth-Vakil linked there, and the corrigendum and further comments at the bottom of Vakil's page). It is possible that the Roth-Vakil example also works for my question, but I don't quite see how to prove it.