Monoidal categorification conjecture for quantum cluster algebras of double Bott–Samelson cells

Let $C$ be the Cartan matrix under consideration, let $\dot{\mathbf f t}$ denote the seeds associated with double Bott–Samelson cells, and let $\overline{\mathcal{U}}(\dot{\mathbf f t})$ be the associated quantum cluster algebra with common triangular basis ${{\mathbf L}}$.

Monoidal categorification conjecture. If $C$ is symmetric, the pair

$$\bigl(\overline{\mathcal{U}}(\dot{\mathbf f t}),{{\mathbf L}}\bigr)$$

admits a monoidal categorification.

A monoidal categorification would explain the cluster algebra and its distinguished basis through a monoidal category, extending the representation-theoretic structures associated with quantum groups. The source does not provide a resolution of this conjecture.