Do we have a moral obligation to genetically cure embryos rather than selecting between them? Such an obligation would be an ethical argument for human germline gene editing (hGGE) to avoid the inheritance of genetic conditions instead of using pre-implantation genetic diagnosis (PGD). In this article, the intuition that we do have such a moral obligation is critically evaluated. The article first develops a theoretical framework for discussing the ethical questions of hGGE. This framework is based on an exploration of the phenomenology of the germline, from both biological and philosophical points of view. It interprets the germline as an embodied intergenerational relationship that carries meanings for the parents and for the children-to-be. It relates them to previous family generations, and to their own children. Hence, the germline is a phenomenologically much richer concept than just the line of cells that carry the inheritable genetic information. Against this background, selection is compared with editing and a key moral difference is identified: editing is in effect an act of co-constructing the genome, which necessarily assumes a wider range of responsibilities that include those parts that are left unedited. Introducing hGGE into societies would hence significantly affect and change the moral structure of the intergenerational relationships. Selective implantation, on the other hand (in the context of PGD), is based on a moral choice in favour of the embryo which is to be unaffected by a disease or disability that causes suffering, rather than selecting knowingly the affected one. The claim that hGGE is in the best interests of the child-to-be counterfactually assumes the presence of a patient who has an interest in being cured. The embryo (a potential future patient) is, however, brought into existence by the same act that is also the treatment. The future children who would result from treatment by hGGE may rather have an interest in not having been treated by hGGE, since it makes the intergenerational relationships more complicated and burdensome. The question 'Is hGGE justified, or even an obligation?' is answered with a No.