The quantum cluster structure conjecture for bosonic extension subalgebras

Let $C$ be a Cartan matrix satisfying $c_{ij}c_{ji}\leq 2$ for all $i\neq j$, and let $b\in\operatorname{Br}^+$ have reduced expression $\mathbf{i}=(i_1,\ldots,i_{\ell(b)})$. Let $\widehat{D}_{\mathbf{i}}[s,\ell(b)]$ be the associated quantum minors, and let $\Lambda$, $B_{\mathrm{GLS}}$, and $K^{\mathrm{ex}}$ be the matrices and exchange set defined from this reduced expression. The initial seed is

$$\mathbf{t}_{\mathbf{i}}:=\{(\widehat{D}_{\mathbf{i}}[s,\ell(b)])_{s\in[1,\ell(b)]},\Lambda,B_{\mathrm{GLS}},K^{\mathrm{ex}}\}.$$

Quantum cluster structure conjecture. The algebra $\widehat{\mathcal{A}}(b)$ is a quantum cluster algebra with initial seed $\mathbf{t}_{\mathbf{i}}$, and its cluster monomials are contained in the global basis $\mathbf{B}(b)$.

This conjecture gives a specific quantum cluster structure on the subalgebra associated with a positive braid element and asserts compatibility of its cluster monomials with the global basis. The supplied text does not state a resolution, so its status remains open.