The fruitfulness conjecture for ordered $Q_n$ quivers and their surface gluings

Let $(Q_n)_{n\in\mathbb{N}}$ be the family of ordered ST-compatible quivers describing the cluster structure for decorated representations into the split real groups $\mathrm{Spin}(n,n+1)$, with the ordered structure described in the paper. Also consider quivers obtained by gluing these quivers according to an ideal triangulation of a bordered surface.

The ordered-quiver fruitfulness conjecture. The quivers $Q_n$ provide fruitful seeds, and any quiver obtained by gluing them according to an ideal triangulation of a bordered surface also provides a fruitful seed.

The case of $Q_2$ is proved because its associated polygonal cluster algebra is of finite type and all mutations are admissible. The conjecture asks for the corresponding fruitfulness statement for the full family and its surface gluings.