Relationship between k-nested automata and Σk+1 plane-walking automata

Ascertain the precise relationship between languages recognized by k-nested tree-walking or plane-walking automata and subshifts recognized by Σk+1 plane-walking automata by proving inclusion(s), equivalence, or separations.

Background

The paper discusses k-nested tree-walking automata, which form a strict hierarchy, and suggests a connection between the k-nested model and the Σk+1 plane-walking automata classes.

However, the authors explicitly note that they currently lack proofs of any formal relationship (inclusion or equivalence) between these classes, making this a concrete research direction.

References

The class of (tree-walking or plane-walking) k-nested automata seems to be related to Σk+1 automata, although we do not have a proof of either direction.

— Subshifts defined by nondeterministic and alternating plane-walking automata (2409.08024 - Menibus et al., 12 Sep 2024) in Section “Strict hierarchy and tree-walking automata”