The canonical isomorphism and quantum deformation conjecture for the double

Let ${\cal D}$ be the cluster double, with Langlands dual ${\cal D}^{\vee}$, and let ${\mathbb Z}_+\{S\}$ denote the semigroup of finitely supported nonnegative-integer-valued functions on a set $S$. Let ${\Bbb L}_+({\cal D}^{\vee})$ be the space of regular positive functions on ${\cal D}^{\vee}$, and let ${\Bbb L}_+({\cal D}_q^{\vee})$ be its quantum analogue. Double duality conjecture. There exists a canonical isomorphism

$${\Bbb I}_{\cal D}:{\mathbb Z}_+\{{\cal D}({\mathbb Z}^t)\}\stackrel{\sim}{\longrightarrow}{\Bbb L}_+({\cal D}^{\vee})$$

compatible with the corresponding maps for the cluster ${\cal A}$- and ${\cal X}$-varieties, and it admits a $q$-deformation

$${\Bbb I}^q_{\cal D}:{\cal D}({\mathbb Z}^t)\stackrel{\sim}{\longrightarrow}{\Bbb L}_+({\cal D}_q^{\vee}).$$

This gives a precise form of the proposed canonical basis and its quantum deformation for the double; the isomorphisms are not proved in the paper.