  In one of his recent papers \cite{ML1}, Moln\'ar showed that if $\mathcal{A}$
is a von Neumann algebra without $I_1, I_2$-type direct summands, then any
function from the positive definite cone of $\mathcal{A}$ to the positive real
numbers preserving the Kubo-Ando power mean for some $0 \neq p \in (-1,1)$ is
necessarily constant. It was shown in that paper, that $I_1$-type algebras
admit nontrivial $p$-power mean preserving functionals, and it was conjectured,
that $I_2$-type algebras admit only constant $p$-power mean preserving
functionals. We confirm the latter. A similar result occurred in another recent
paper of Moln\'ar \cite{ML2} concerning the Wasserstein mean. We prove the
conjecture for $I_2$-type algebras in regard of the Wasserstein mean, too. We
also give two conditions that characterise centrality in $C^*$-algebras.
