  In \cite{molnar} L.~Molnar studied the question of whether the L\"owner
partial order on the positive cone of an operator algebra is determined by the
norm of any arbitrary Kubo-Ando mean. He affirmatively answered the question
for certain classes of Kubo-Ando means and left as an open problem the general
case. We here give an answer to this question, by showing that the norm of
every symmetric Kubo-Ando mean $\sigma$ on $\mathcal B(H)$ is
order-determining, i.e. if $A, B\in \mathcal B(H)^{{\sss{++}}}$ satisfy $\Vert
A\sigma X\Vert \le \Vert B\sigma X\Vert$ for every $X\in \mathcal
B(H)^{\sss{++}}$, then $A\le B$.
