The Burnside semigroup stability conjecture

Let

$$\mathcal{B}(m,n)=\langle A\mid t^m=t^{m+n}\quad\forall t\in A^+\rangle$$

be a Burnside semigroup, and for $s\in\mathcal{B}(m,n)$ let $S^{\mathscr{J}}_s$ be the finite semigroup formed by the elements $\mathscr{J}$-above $s$, together with zero $\square$. Let $\mathsf{KR}(S^{\mathscr{J}}_s,A)$ be its Karnofsky–Rhodes expansion. Burnside stability conjecture. The pair $(\mathsf{KR}(S^{\mathscr{J}}_s),A)$ satisfies the stability condition $\mathsf{Mc}\circ\mathsf{KR}(S,A)=(S,A)$. The conjecture proposes stability under both the Karnofsky–Rhodes and McCammond expansions for these finite pieces of Burnside semigroups. The supplied text gives no evidence of resolution.