Some sources indicate that not all of the Israelites left Egypt during the Exodus. On the other hand, the initial "conversion" of the Israelites to Judaism occurred at Matan Torah at Mt. Sinai during the Exodus.
Are the Israelites who stayed behind in Egypt and their descendants Jews?
I do recognize that identifying these people and especially their modern-day descendants is an exercise in genealogical futility, but suppose hypothetically that someone were to show up today at a synagogue with essentially conclusive proof that they are fully paternally and maternally descended from members of the Twelve Tribes who did not participate in the Exodus and are not maternally descended from anyone who was present at Matan Torah. What is such a person's halachic status as a Jew?
- Are they a full Jew, eligible to participate immediately on an equal standing with their brethren whose ancestors participated in the Exodus?
- Are they a full Gentile, eligible to become a righteous Noachide or seek conversion to Judaism at their discretion?
- Do they occupy some sort of halfway status, halachically Jewish but ineligible to participate in some or all mitzvot unless they "make up" what their ancestors failed to do at Sinai (e.g. through some sort of "conversion-like" process)?