Local embeddability conjecture for CR 3-spaces

A CR 3-space is a three-dimensional CR manifold, and its canonical bundle is the complex line bundle of top-degree CR forms. A section is closed when its exterior derivative vanishes, and locally embeddable when it is locally realizable as a CR submanifold of a complex manifold.

Local embeddability conjecture. A CR 3-space admits locally a closed, non-zero section of its canonical bundle if and only if it is locally embeddable.

The claim concerns the relationship between local embeddability and the existence of non-zero closed canonical-bundle sections, which encode the relevant Maxwell solutions in the construction discussed in the source. The source notes that analogous examples are known in dimension seven, while the existence of such examples in dimensions three and five is unclear.