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On the locally distinguishable three-dimensional subspace of bipartite systems

It has been conjectured and numerically supported in King and Matysiak (2007 J. Phys. A: Math. Theor. 40 7939) that any three-dimensional subspace of C3⊗Cn has an orthonormal basis that can be reliably distinguished using one-way local operations and classical communications (LOCC). Since then there has been little progress towards the conjecture. We propose and prove a few cases of the conjecture. We also present a few equivalent or sufficient conditions for the conjecture, and thus moderately simplify the conjecture.