Regularity of languages recognized by non-meta finite-state string machines

Prove that every language over the generator alphabet Σ accepted by a string machine composed solely of finite-state deterministic transducers without meta-vertices—whose only free input is a morphism X → X in a tape category generated by endomorphisms of X (with strings identified as endomorphisms that omit the copy morphism) and whose sole output is acceptance via a designated subset of a finite state space—is regular.

Background

The authors paper finite-state deterministic transducers operating over tape categories and analyze their ability to prepend and manipulate inputs. They present a lemma bounding the memory needed for certain compositions and argue that, despite additional compositional features, such machines may still be limited in power.

They put forward a conjecture that any language decided by such a machine is regular, motivated by constant-space decision procedures and classical results (e.g., DSPACE(O(1)) = REG and one-tape Turing machine time lower bounds). A proof would clarify the exact expressive boundary of non-meta finite-state string machines.