Determine the language-theoretic classification of the D-, R-, and L-representative languages

Determine where the representative languages D, R, and L for free inverse monoids reside within the Chomsky hierarchy, including whether they are context-sensitive or belong to language classes such as the indexed languages.

Background

For free inverse monoids of rank at least two, the paper proves that the Green representative languages D, R, and L are neither context-free nor co-context-free. Their precise position among more expressive formal-language classes is not established. The authors explicitly identify determining whether these languages are context-sensitive or indexed as an unresolved problem.