Quantum cluster algebra conjecture for triple intersections

Let $\mathfrak{g}$ be the underlying symmetrizable quantum group, let $w_{1}$ and $w_{2}$ be Weyl group elements, and let $T_{w}$ denote the corresponding braid-group action. Write $\mathbf{U}_q^{\geq 0}$ for the nonnegative part and order Weyl group elements by the Bruhat order. Consider the triple intersection

$$\mathbf{U}_q^{-}\cap T_{w_{1}^{-1}}^{-1}\mathbf{U}_q^{-}\cap T_{w_{2}^{-1}}^{-1}\mathbf{U}_q^{\geq 0}.$$

Quantum cluster intersection conjecture. If $w_{1}\leq w_{2}$ in the Bruhat order, this triple intersection contains a quantum cluster algebra, and its quantum cluster monomials are contained in

$$\mathbf{B}^{\mathrm{up}}\cap\mathbf{U}_q^{-}\cap T_{w_{1}}\mathbf{U}_q^{-}\cap T_{w_{2}}\mathbf{U}_q^{\geq 0},$$

where $\mathbf{B}^{\mathrm{up}}$ is the dual canonical basis.

The surrounding discussion says that parts of the claim appear to be known in simply-laced or symmetric types, but does not establish the full assertion in the stated generality. Thus the conjecture remains open on the supplied evidence.