What happens to jointly held debt?

Agreed. The estate is worked through in four steps, in this order: (1) the cost of preparing, shrouding and burying the body; (2) the deceased’s debts; (3) any bequest, capped at one third of what is left after (1) and (2); and (4) the fixed shares of the heirs, out of the remainder. The Kuwaiti Encyclopaedia of Islamic Jurisprudence records the sequence itself as a point on which there is no disagreement among the jurists, and records debt-before-bequest as agreed by all of them — even though Surah an-Nisa names the bequest before the debt, the jurists read the order as debt first, because a debt is owed from the outset while a bequest is a voluntary gift. Two rankings inside this sequence are genuinely disputed and are researched separately: whether a debt secured on a particular asset outranks the funeral itself, and whether unpaid religious dues such as zakat or an unperformed hajj come out of the estate at all where the deceased left no instruction. Neither dispute disturbs the four-step order. We checked this point; the schools do not differ on it.

Shared evidence: Surah an-Nisa 4:11 — “after the fulfilment of bequests and debts” · Surah an-Nisa 4:12 — the same clause, twice more · Sahih Muslim 1206a — the pilgrim who died at ‘Arafah was washed and shrouded with no inquiry into his debts.