Suppose we have a seed $(x,y,B)$ where $B$ is a skew-symmetrizable matrix, $x = \{x_1,\ldots,x_n\}$ and y is an n-tuple of elements in $Trop\{x_{n+1},\ldots,x_m\}$.

If there are finitely many cluster variables in the cluster algebra generated by this seed, then why are there only finitely many y-variables? (This is equivalent to asking why the extended part of $\tilde{B}$ stays bounded if there are only finitely many cluster variables.) -- it is mentioned here (https://arxiv.org/pdf/1707.07190.pdf) in the proof of Corollary 5.1.6 which makes me think it is trivial.

If there are finitely many cluster variables, it is clear why the mutation equivalence class of $B$ is finite, because otherwise there would be a mutation equivalent matrix $B'$ with an entry $b'_{ij} \geq 4$ and then performing a sequence of mutations at $i$ and $j$ would generate infinitely many cluster variables.