Uniqueness of simultaneously stabilizable and stabilized means

Let $K$, $M$, and $N$ be means, and let $G$ denote the geometric mean. For means $A$ and $B$, write $A\otimes B$ for their indicated mean composition. Uniqueness conjecture for stabilizable and stabilized means.

If mean $N$ is simultaneously $(K,M)$- and $(M,K)$-stabilizable, then $N=K=M$. If mean $M$ is simultaneously $(K,N)$- and $(N,K)$-stabilized, then either $M=K=N$ or $M=G=K\otimes N$. If mean $M$ is simultaneously $(K,N)$-stabilizable and $(K,N)$-stabilized, then $M=K=N$.

These assertions summarize consequences suggested by the asymptotic equalities developed in the paper. The supplied text does not state whether they have been proved or remain open.