Further compact nearly Kähler six-manifolds

Determine whether there exist additional compact nearly Kähler six-manifolds beyond the currently known examples.

Background

The homogeneous nearly Kähler six-manifolds are completely classified (S6, F1,2(C3), CP3, and S3×S3). Foscolo–Haskins constructed the first inhomogeneous compact examples on S6 and S3×S3. The notes state that, at present, no other compact nearly Kähler six-manifolds are known.

This open existence question probes whether the current list is exhaustive or if new compact geometries (potentially on different underlying manifolds or with different symmetry types) can be constructed.