Commutativity from the Equality of two Heron-type Means in $C^{\ast }$-Algebras

Let $\mathcal{A}$ be a unital $C^{\ast }$-algebra, and let ${\mathcal{A}}^{++}$ denote the cone of positive invertible elements.We prove that for $A,B\in {\mathcal{A}}^{++}$, the equality between the conventional Heron-type mean $$(\frac{A^{1\mathrm{/}2}+B^{1\mathrm{/}2}}{2}{)}^2$$ and the Wasserstein mean $$\frac{1}{4}(A+B+A(A^{-1}\mathrm{\#}B)+(A^{-1}\mathrm{\#}B)A)$$ forces $A$ and $B$ to commute, thereby answering \cite[Problem~1]{MS24} posed by Molnár and Simon.Our proof does not require any tracial functional; instead it relies on a characterization of the operator-valued triangle equality due to Ando and Hayashi.