In our very recent fair division paper, we prove that PROP1 allocations exist and can be computed efficiently on instances with (not necessarily monotone) submodular and subadditive valuations. Among other results, we show that maximum Nash welfare allocations are PPOP1 on submodular instances.

Martin Jupakkal Andersen, Ioannis Caragiannis, Anders Bo Ipsen, Alexander S{\o}ltoft Computing Approximately Proportional Allocations of Indivisible Goods: Beyond Additive and Monotone Valuations https://arxiv.org/abs/2508.12453