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In Horodecki, Horodecki and Horodecki (1998), Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?, the authors remark in the conclusions (beginning of pag. 4, emphasis mine):

So, one is now faced with the problem similar to the initial one, i.e., are the inseparable states with positive partial transposition nonlocal?

In other words, the question is whether inseparable states which have a positive partial transpose can (or must?) exhibit nonlocal correlations.

This was an open question in 1998. Has it been answered in the meantime, or were there developments in this direction?

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This question was solved in 2014 by Vértesi and Brunner: they found a quantum state with positive partial transposition that violated a Bell inequality. The conjecture that all states with positive partial transposition do not violate any Bell inequality was known as the Peres conjecture, so they disproved it.

As for your parenthetical question, whether all states with positive partial transposition do violate a Bell inequality, the answer is negative. It has been known since 1989 that there exists entangled states that do not violate any Bell inequality; more recently it was shown that there exists states that still do not violate any Bell inequality even after local filtering. Curiously, both these examples are states with negative partial transposition, but it should be easy enough to construct one with positive partial transposition.

EDIT: It turns out that it is not easy, but it has been done. In this paper they showed a PPT entangled state that is local for all projective measurements, and in this paper a tripartite PPT entangled state that is local for all POVMs.

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