The cardinality conjecture for automorphism groups of superextensions of monogenic semigroups

Let $\mathrm{M}_{r,m}$ denote a finite monogenic semigroup, let $\lambda(S)$ denote its superextension, and let $\operatorname{Aut}(G)$ denote the automorphism group of a group $G$. For integers $r,s\geq 3$ and $n,m\geq 1$ satisfying $r+m=s+n$, consider the automorphism groups of the superextensions associated with $\mathrm{M}_{r,m}$ and $\mathrm{M}_{s,n}$. Cardinality conjecture. The groups

$$\operatorname{Aut}(\lambda(\mathrm{M}_{r,m}))$$

and

$$\operatorname{Aut}(\lambda(\mathrm{M}_{s,n}))$$

are isomorphic. This predicts that the automorphism group depends on the common value of $r+m=s+n$ for the stated range of parameters. The conjecture is based on the computed cases of cardinality at most $5$; no resolution is supplied in the source.