Decidability of HD0L ω-equivalence in the general case

Determine whether the HD0L ω-equivalence problem is decidable in the general case: given two morphisms f: Γ_f → Γ_f^+ and g: Γ_g → Γ_g^+ over finite alphabets, distinguished start symbols, and codings τ: Γ_f → Σ and ρ: Γ_g → Σ, decide whether the morphic sequences τ(f^∞(start_f)) and ρ(g^∞(start_g)) are equal, without assuming primitivity or other restrictions on the morphisms.

Background

The paper develops an elementary, automatable method (Theorem 4.1) to prove equality of morphic sequences represented by different morphisms and codings, including successful automatic proofs for subsequences of the Fibonacci sequence. Despite these successes, the authors note a limited understanding of the overall power of the approach.

They specifically highlight the broader decision problem of whether two given HD0L representations (morphism plus coding) generate the same infinite sequence. This problem, known as HD0L ω-equivalence, is known to be decidable for primitive morphisms due to results by Durand (2012), but the decidability status in the unrestricted, general case remains unresolved.