Non-emptiness of unlabeled regions in the hierarchy diagram

Determine whether the currently unlabeled set-theoretic regions in the inclusion diagram among the classes 2CrevLIN (languages accepted by complete reversible 2-head automata), 2rev1LIN (languages accepted by 1-limited reversible 2-head automata), 2revLIN (languages accepted by reversible deterministic 2-head automata), 2detLIN (languages accepted by deterministic 2-head automata), and LDLL (languages generated by left deterministic linear grammars) are non-empty by constructing explicit separating example languages, or prove that some of these regions are empty.

Background

The paper provides a diagram (Fig. 1) summarizing proven inclusions and separations among five language families: 2detLIN, 2revLIN, 2rev1LIN, 2CrevLIN, and LDLL. While several separations are established with example languages, the authors indicate there remain regions in the diagram without witnessing examples.

They state explicitly that finding examples to populate these unlabeled regions or proving such regions empty constitutes part of the open problems. Addressing this will sharpen the structural understanding of the expressive power of reversible two-head automata vis-à-vis deterministic two-head automata and left deterministic linear grammars.