The combinatorial principal-coefficients exchange-graph conjecture

Let $\mathbb{T}_n$ be the $n$-regular tree, and let $(\tilde B_t)_{t\in\mathbb{T}_n}$ be the family of $2n\times n$ extended exchange matrices for the principal-coefficients cluster algebra $\mathcal{A}_\bullet(t_0)$. Write $B_t$ for the principal part of $\tilde B_t$, and call $(\mathbf{y},B)$ a $Y$-seed. Combinatorial principal-coefficients exchange-graph conjecture. Two labeled seeds at vertices $t,t'$ define the same seed of $\mathcal{A}_\bullet(t_0)$ if and only if $\tilde B_{t'}$ is obtained from $\tilde B_t$ by simultaneously permuting the rows and columns belonging to the principal part $B_t$; equivalently, the cluster of a seed $(\mathbf{x},\mathbf{y},B)$ is uniquely determined by its $Y$-seed $(\mathbf{y},B)$. The conjecture gives a purely matrix-mutational description of the principal-coefficients exchange graph and was not yet checked even for finite type in the paper.