Do Denominator Vectors Determine the Cluster Variable

Question

Given a cluster algebra $A=A(\mathbf{x},Q)$, the Laurent Phenomenon states that all the cluster variables of $A$ are Laurent polynomials in the elements of $\mathbf{x}$. Thus, any cluster variable $y$ can be written $$y=\frac{p(x_1,\dots,x_n)}{x_1^{d_1}\cdots x_n^{d_n}}$$ where $p$ is a polynomial and $d_i$ are positive integers. We call $d(y):=(d_1,\dots,d_n)$ the denominator vector of $y$.

If $Q$ is mutation equivalent to a simply laced Dynkin diagram, all cluster variables are uniquely determined by their denominator vector. I would like to know to what extent this holds in general. That is:

Is it true that for any cluster algebra, the clusters are determined by their denominator vectors? If not, what classes of cluster algebras have this property? I am particularly interested in surface cluster algebras.

Answer 1

For acyclic initial seeds, this is now Corollary 8 of Rupel and Stella arXiv:1712.08478 .

Answer 2

I believe this is conjectured to be true in general.

As for cluster algebras from surfaces, maybe the answer lies in http://math.berkeley.edu/~williams/papers/MSW-July24v5.pdf ?

Answer 3

For acyclic cluster algebras, a theorem like that can be found in "Buan, Marsh, Reiten: Denominators of cluster variables, J. London Math. Soc. 2009", Theorem 1.3 (preprint).