Deterministic class as intersection of existential and universal classes

Determine whether, for each n ≥ 1, the deterministic class Δn of subshifts recognized by plane-walking automata with at most n−1 quantifier blocks equals the intersection Σn ∩ Πn.

Background

After introducing the Σn, Πn, and Δn classes via quantifier-block structure on paths of plane-walking automata, the authors note the evident inclusions Δn ⊆ Σn ⊆ Δn+1 and Δn ⊆ Πn ⊆ Δn+1 but highlight uncertainty about whether Δn coincides with the intersection Σn ∩ Πn.

Resolving this would clarify whether simultaneous existential and universal acceptance at a fixed alternation depth can always be simulated by a deterministic automaton at that depth.