Is every entangled state nonlocal given many copies?
In plain words
Some entangled states never violate a Bell inequality when measured one pair at a time. The question is whether measuring many copies together, with local filtering, always reveals nonlocality for every entangled state.
Precise statement
For an entangled bipartite state $\rho$, determine whether some $k$ and local measurements on $k$ copies of $\rho$ (possibly after local filtering, with no auxiliary entangled states and no extra shared entanglement) violate a Bell inequality. Superactivation for some states was shown by Palazuelos (2012) and Bell violation by bound-entangled states by Vertesi and Brunner (2014). An answer is a proof for all entangled states or an entangled state with a local model for every $k$.
What would settle it
A general activation proof, or a family of local models covering all numbers of copies.
Status in the literature
With auxiliary entangled states in a network, every entangled state can be certified device-independently (Bowles, Supic, Cavalcanti, Acin, PRL 2018, doi:10.1103/PhysRevLett.121.180503); the many-copy question without such help is open.