Title: A conjecture on $C$-matrices of cluster algebras,
Abstract: For a skew-symmetrizable cluster algebra $\mathcal A_{t_0}$ with principal
coefficients at $t_0$, we prove that each seed $\Sigma_t$ of $\mathcal A_{t_0}$
is uniquely determined by its {\bf C-matrix}, which was proposed by Fomin and
Zelevinsky in \cite{FZ3} as a conjecture. Our proof is based on the fact that
the positivity of cluster variables and sign-coherence of $c$-vectors hold for
$\mathcal A_{t_0}$, which was actually verified in \cite{GHKK}. More discussion
is given in the sign-skew-symmetric case so as to obtain a conclusion as weak
version of the conjecture in this general case.