Let $ \mathcal A$ be the complex cluster algebra obtained from the initial seed $(a,x, B)$ where : $x$ stands for a $n$-tuple $(x_1,\ldots,x_n)$ of complex indeterminates, idem for $a=(a_1,\ldots,a_n) $ and $B=(b_{ij})_{i,j=1}^n$ is the associated exchange matrix).
Definition: The "$X$-mutation in the direction $k$" is the birational map
$$x\mapsto \mu_k(x)=x'=(x'_i)_{i=1}^n$$
where
$$ x_k'=\frac{1}{x_k} \; \quad \mbox{ and } \quad
x_i'=x_i\left( 1+x_k^{-[b_{ik}]_+}
\right)^{-b_{ik}} \qquad \mbox{ for any }\quad i\neq k
$$
(with $[s]_+=\max(0,s)$ for any $s\in \mathbb R$).
Definition: The "$A$-mutation in the direction $k$" is the birational map
$$a\mapsto \mu_k(a)=a'=(a'_i)_{i=1}^n$$ where
$$ a_i'=a_i \; \mbox{ if }\, i\neq k \quad \mbox{ and } \quad
a_k'=\frac{1}{a_k}\left(
\prod_{j \vert b_{kj} > 0} a_j^{b_{kj}} + \prod_{j \vert b_{kj} < 0} a_j^{-b_{kj}}
\right) \, .
$$
(Cf. $\S$1.2.4 in V. Fock - A. Goncharov. Cluster ensembles, quantization and the dilogarithm. Ann. ÉNS (2009).)
By definition, the associated $X$-cluster variables (resp. $A$-cluster variables) (in Fock-Goncharov's terminology) are all the rational functions $x'=x'(x_1,\ldots,x_n)$ (resp. $a'=a'(a_1,\ldots,a_n)$) which can be obtained from the initial $x_i$'s (resp. from the $a_i$'s) by finite sequences of $X$-mutations (resp. of $A$-mutations) (note that the exchange matrix B mutates at each step as well but it is pointless to give explicit formula for the matrix mutations $B\mapsto \mu_k(B)$ here).
The following question mainly concerns the $X$-cluster variables :
Question: is it known whether the following statement holds true or not :
given two X-cluster variables $x$ and $x'$, one has : $$ (\mathcal S)\qquad \quad dx \wedge dx' \equiv 0 \qquad \mbox{ if and only if }\qquad x'=x \; \mbox{ or } \; x'=1/x\; \mbox{ ?} $$
(Note : here "$dx$" stands for the total derivative of $x$ and " $\wedge$ " for the usual wedge product of differential algebra; moreover " $x'=x$ or $x'=1/x$ " has to be understood as an equality between two complex rational functions in the $x_i$'s).
1). If it is true and well known, a reference would be welcome;
2). If ($\mathcal S$) does not hold true in full generality, what about the case of cluster algebras of finite type (with or without frozen variables)?
3). More generally, if ($\mathcal S$) does not hold true in full generality, are there some "natural and nice" conditions on $\mathcal A$ (i.e. on $B$) ensuring that ($\mathcal S$) is satisfied?
4). (Not of primary interest): same questions but for the $A$-cluster variables.
Remark: looking at many explicit cases, I'm pretty sure that $(\mathcal S)$ indeed holds true in case 2) (more precisely : for finite type without frozen variables). This must be well-known by peoples familiar with the theory of cluster algebras... but I'm not such a person!
Any help would be welcome. Thanks in advances!
