Exchange-graph isomorphism and seed-exchange-matrix reconstruction conjecture

Let $d4d0$ and $d4d0'$ be connected cluster algebras, each of finite type or skew-symmetric finite mutation type. Let $d4e2=(\mathbf{x},B)$ and $d4e2'=(\mathbf{x}',B')$ be seeds of $d4d0$ and $d4d0'$, respectively. For a seed $d4e2$, let $N(\ell^k_{d4e2})$ denote the associated quantity appearing in the source, for $k\in\mathbb{Z}^{*}$. Exchange-graph and matrix conjecture. If $N(\ell^k_{d4e2})=N(\ell^k_{d4e2'})$ for every $k\in\mathbb{Z}^{*}$, then there is an isomorphism

$$\sigma:E_\mathcal{A}\to E_{\mathcal{A}'}$$

satisfying $d4d7(d4e2)=d4e2'$. Moreover, assuming that neither $d4d0$ nor $d4d0'$ is of rank $2$ or type $F_4$, the same equality implies

$$B\cong B'\quad\text{or}\quad B\cong -B'.$$

The claim relates numerical data associated with loops in exchange graphs to their global isomorphism type and, except for the stated rank-$2$ and type-$F_4$ cases, to the exchange matrices of corresponding seeds. The supplied parser marks its resolution status as unknown, so the database records it as open.