Our review of the scientific literature identified no studies directly addressing the cryopreservation of a pregnant individual or the preservation of an embryo or fetus in utero, despite the routine use of cryopreservation for embryos, oocytes, and sperm in reproductive medicine and the enrollment of individuals of reproductive age in biostasis organizations. This paper defines this unexamined problem by synthesizing evidence from cryobiology, reproductive and developmental biology, clinical medicine, bioethics, and law. It develops a conceptual framework that treats the pregnant individual and conceptus as one physical body containing two biologically distinct systems, and distinguishes established evidence, testable hypotheses, and speculative scenarios. Two falsifiable hypotheses follow: that cryoprotectant delivery across the maternal-fetal interface is rate-limited under whole-body perfusion, and that the resulting maternal-fetal concentration gradient varies with gestational age. The paper analyzes four preservation scenarios, the clinical precedent of somatic support after maternal brain death, and the consent, surrogate decision-making, and jurisdictional questions these scenarios raise. Membership data from three biostasis organizations indicate that roughly one in eight members of organizations reporting age data is a woman of reproductive age. The paper does not propose clinical protocols or claim feasibility. It sets out a prioritized research agenda and recommendations for organizational policy in a previously unexplored area of biomedical research.