Enomoto–Kashiwara's global crystal-basis conjecture for $V_\theta(\lambda)$

Let $V_\theta(\lambda)$ be the symmetric-crystal representation for a dominant integral weight $\lambda$ fixed by $\theta$, and let $L_\theta(\lambda)$ be its crystal lattice with bar-conjugate lattice $L_\theta(\lambda)^-$. Let $V_\theta(\lambda)^{\mathrm{low}}_\mathbf A=U_{\mathbf A}\,v_\lambda$ be the specified integral form. A triple of lattices is balanced when it has the balancing property defined in the paper. Enomoto–Kashiwara's global crystal-basis conjecture. The representation $V_\theta(\lambda)$ has a global crystal basis; equivalently, the triple

$$\bigl(L_\theta(\lambda),L_\theta(\lambda)^-,V_\theta(\lambda)^{\mathrm{low}}_\mathbf A\bigr)$$

is balanced. This is the global, integral refinement of the preceding crystal-basis conjecture.