20.12 (2022)

Let us say that a group $G$ has the unique $n$-fold product property (u.-n-p.) if for every $n$-tuple of finite nonempty subsets $A_1, \dots, A_n \subseteq G$ there exists $g \in G$ which can be written in one and only one way as $g = a_1 \dots a_n$ with $a_i \in A_i$. It is easy to see that u.-n-p. implies u.-m-p. for $n \geqslant m$ (since some of the $A_i$ can be $\{1\}$).

Are the conditions u.-n-p. ($n \geqslant 2$) all equivalent to u.-2-p., the usual unique product condition?