Global signed-crystal structure conjecture for the quantum alcove model

Let $\lambda$ be an arbitrary weight and let $\Gamma$ be a reduced chain of roots corresponding to $\lambda$. Write $\mathcal{A}(\Gamma)$ for the associated quantum alcove model, and suppose it carries the partial signed crystal structure constructed in the source. A generalized quantum Yang--Baxter move is defined by a sijection $(I_1,I_2,Y)$. Global signed-crystal structure conjecture. For every reduced chain of roots $\Gamma$ corresponding to an arbitrary weight $\lambda$, there exists a signed crystal structure on the whole of $\mathcal{A}(\Gamma)$ extending the partial crystal structure defined above. Moreover, the sijection $(I_1,I_2,Y)$ defining a generalized quantum Yang--Baxter move in Theorem commutes with the crystal operators defined on $\mathcal{A}(\Gamma)$. This would complete the partial construction and make generalized quantum Yang--Baxter moves compatible with the resulting crystal structure, but the source gives no resolution of the conjecture.