Möbius Problem

Let $A=\{a_1, a_2, \ldots\}$ be a free Abelian Semigroup, where $a_1$ is the unit element, and let $\mu(n)$ be the Möbius Function. Then do the following properties,

1. $a<b$ Implies $ac<bc$ for $a,b,c\in A$, where $A$ has the linear order $a_1<a_2<\ldots$,

2. $\mu(a_n)=\mu(n)$ for all $n$,

imply that

$$a_{mn}=a_m a_n$$

for all $m,n\geq 1$? The problem is known to be true for $mn\leq 74$ for all $n\leq 240$.

See also Braun's Conjecture, Möbius Function

References

Flath, A. and Zulauf, A. ``Does the Möbius Function Determine Multiplicative Arithmetic?'' Amer. Math. Monthly 102, 354-256, 1995.