The NPT bound-entanglement conjecture

Let $\rho$ be a bipartite state. It is NPT when its partial transpose is not positive semidefinite, and it is distillable when some tensor power admits a Schmidt-rank-at-most-two vector with negative expectation under the corresponding partial transpose.

NPT bound-entanglement conjecture. There exist bipartite NPT states which are not distillable.

Every distillable state is NPT, but it is unknown whether every NPT state is distillable. The conjecture asserts that the converse fails and would provide NPT bound entanglement.