Partition rank of the four-party state with unresolved rank-two certification

Determine whether the three-qutrit state $|\psi(3,4,5)\rangle$ admits a partition-rank-two decomposition.

Background

The appendix develops a criterion based on Schmidt coefficients that can rule out orthogonal partition-rank decompositions with specified partition-rank vectors. The criterion excludes partition-rank-two decompositions for several states discussed in the main text, including the GHZ state, the supersinglet, and $|\psi(1,1,1)\rangle$.

For $|\psi(3,4,5)\rangle$, however, the criterion does not settle whether a partition-rank-two decomposition exists. Thus the existence or nonexistence of such a decomposition remains unresolved in the stated discussion.