The Fock–Goncharov basis conjecture for upper cluster algebras

Let $\mathcal{U}$ be the upper cluster algebra, and let $\mathcal{M}^{\circ}$ be the set of tropical points, obtained as equivalence classes of lattice points under tropical transformations between seeds. A set is parametrized by the tropical points if it has one element indexed by each tropical point and the element indexed by $[g]$ is $[g]$-pointed in every seed.

Fock–Goncharov conjecture. The upper cluster algebra $\mathcal{U}$ possesses a basis parametrized by the tropical points.

The conjecture seeks a canonical basis indexed by tropical data; the source also records the further expectation that this basis is positive, meaning that its structure constants are non-negative. The supplied source gives no resolution status.