Generalized-minor realization of regular cluster variables for Coxeter double Bruhat cells

Let $Q$ be a finite valued quiver without oriented cycles, and let $G^{c,c^{-1}} \subset G$ be the associated Coxeter double Bruhat cell. The algebra $\mathcal{A}_{\widetilde{Q}_{dp}}$ is identified with $\Bbbk[G^{c,c^{-1}}]$. A generalized minor is a matrix coefficient associated with a representation and weights; a cluster variable is regular if it arises from a regular representation. Generalized-minor realization conjecture. All regular cluster variables in $\mathcal{A}_{\widetilde{Q}_{dp}}$ are generalized minors of representations that are neither highest- nor lowest-weight, and the weight of each such minor is the $\mathbf{g}$-vector of the cluster variable. This would extend the generalized-minor description beyond preprojective and postinjective cluster variables, whose defining relations come from global generalized determinantal identities, to level zero and regular representations; an interpretation of their exchange relations remains desirable and is not provided here.