  We derive constraints on Lagrangian embeddings in completions of certain
stable symplectic fillings with semisimple symplectic cohomologies. Manifolds
with these properties can be constructed by generalizing the boundary connected
sum operation to our setting, and are related to certain birational surgeries
like blow-downs and flips. As a consequence, there are many non-toric
(non-compact) monotone symplectic manifolds whose wrapped Fukaya categories are
proper.
