Product groups Σ×Λ are not very flexibly P-stable when Σ admits a non-abelian free quotient and Λ lacks property (τ), so P-stability is not closed under direct products.
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Stability for product groups and property $(\tau)$
Product groups Σ×Λ are not very flexibly P-stable when Σ admits a non-abelian free quotient and Λ lacks property (τ), so P-stability is not closed under direct products.