Regularity conjecture for finite-state string machines without a meta-vertex

Let a string machine be composed only of finite-state transducers, with the input and output categories of each transducer being tape categories. Suppose it has one free input $X\to X$ in a tape category $\mathcal{X}$ whose generating morphisms form a set $\Sigma$ and are all endomorphisms of the generating object $X$. Assume its sole output is a state space $S(A)$, with some subset designated as accepting. An endomorphism $X\to X$ in $\mathcal{X}$ that contains no instances of the copy morphism is regarded as a string over $\Sigma$. Regularity conjecture. Every language over $\Sigma$ accepted by this string machine is regular. The conjecture asserts that finite-state string machines without a meta-vertex have no more language-recognition power than finite automata; the paper presents a sketch intended to establish it, but the supplied text gives no resolution status.