Preservers of the $p$-power and the Wasserstein means on $2\times 2$ matrices

In one of his recent papers \cite{ML1}, Molnár 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 $0\mathrm{\ne }p\in (-1,1)$ 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 Molnár \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 $C^{\ast }$-algebras.