The period-two formula for the affine-quiver quantity $\tilde{K}_n$ in type $\tilde{E}_8$

Let $a_n$ be the cluster variables associated with the extending vertex, and let $\tilde{K}_n$ denote the quantity whose period-two behavior is being conjectured. Period-two formula for $\tilde{K}_n$. The expression

$$\tilde{K}_n:=\frac{a_{n+30}+a_n}{a_{n+15}}$$

is period $2$. The source states that this conjecture was not proved and supplies it as an ansatz to complete the corresponding $\tilde{E}_8$ period table. The notation $\tilde{K}_n$ is reused from the preceding discussion, while this formula concerns the $\tilde{E}_8$ relation.