The Kleene-expression probability conjecture for Karnofsky–Rhodes expansions

Let $S$ be a finite semigroup with generating set $A$, let $(\mathsf{KR}(S,A),A)$ be its Karnofsky–Rhodes expansion, and let $K(\mathsf{KR}(S,A))$ be a Kleene expression for its normal forms. Write $\Psi^{\mathsf{KR}(S,A)}_w$ for the corresponding weights. Kleene-expression probability conjecture. The Kleene expressions constructed in the cited work satisfy

$$\sum_{w\in K(\mathsf{KR}(S,A))} \Psi^{\mathsf{KR}(S,A)}_w = 1.$$

This condition ensures that every normal form occurs exactly once in the constructed Kleene expression, allowing the stationary distribution to be computed from rational expressions. The statement is presented without evidence of resolution in the supplied text.