The trivial-range conjecture for restriction automorphisms of superextensions

Let $\mathrm{M}_{r,m}$ be a finite monogenic semigroup, let $\mathrm{M}_{r,m}^{\cdot 2}$ denote the corresponding semigroup appearing in the restriction, let $\lambda(S)$ denote the superextension of a semigroup $S$, and let $\operatorname{Aut}(G)$ denote the automorphism group of a group $G$. For $r\geq 2$, $m\geq 1$, define the restriction operator

$$R:\operatorname{Aut}(\lambda(\mathrm{M}_{r,m}))\to\operatorname{Aut}(\lambda(\mathrm{M}_{r,m}^{\cdot 2})),\qquad R(\psi)=\psi|_{\lambda(\mathrm{M}_{r,m}^{\cdot 2})}.$$

Trivial-range conjecture. The restriction operator $R$ has trivial range. The conjecture concerns the restriction of automorphisms from the superextension of $\mathrm{M}_{r,m}$ to that of $\mathrm{M}_{r,m}^{\cdot 2}$; the source gives no resolution beyond the computed examples.