Dual global-basis balancedness conjecture for the symmetric-crystal module

Let $L_\theta(\lambda)$ be the crystal lattice, let $c(L_\theta(\lambda))$ denote its bar-conjugate lattice, and let $V_\theta(\lambda)^{\mathrm{up}}_{\mathbf A}$ be the dual lattice with respect to the invariant bilinear form. Dual balancedness conjecture. The triple

$$\left(L_\theta(\lambda),c(L_\theta(\lambda)),V_\theta(\lambda)^{\mathrm{up}}_{\mathbf A}\right)$$

is balanced. The source states this as an equivalent dual formulation of the global-basis conjecture, so it carries the same significance and unresolved status.