Supermoduli spaces are non-projected in every genus $g\ge 3$

We prove that, for every genus $g\ge 3$, the primary obstruction of each parity component of the moduli superstack of smooth unpunctured super Riemann surfaces is nonzero, both algebraically and holomorphically. Consequently, both components are non-projected and, in particular, non-split: neither admits an algebraic or holomorphic retraction onto its reduced moduli stack of spin curves.