The stability criterion for McCammond expansions

Let $(S,A)$ be a finite semigroup with generating set $A$, and let $\mathsf{KR}(S,A)$ and $\mathsf{Mc}$ denote the Karnofsky–Rhodes and McCammond expansions, respectively. The expansion is stable under $\mathsf{Mc}$ when $\mathsf{Mc}\circ\mathsf{KR}(S,A)=\mathsf{KR}(S,A)$. Stability criterion conjecture. $\mathsf{KR}(S,A)$ is stable under $\mathsf{Mc}$ if and only if $\mathsf{Mc}\circ\mathsf{KR}(S,A)$ is a right Cayley graph; equivalently, if $\mathsf{Mc}$ changes any of the $\mathscr{R}$-classes of $\mathsf{KR}(S,A)$, then $\mathsf{Mc}\circ\mathsf{KR}(S,A)$ cannot be a right Cayley graph. This criterion relates the expansion's stability to whether the resulting graph retains the structure of a right Cayley graph. No resolution is given in the supplied text.