Conjecture on Demazure crystal expansions of generalized minors

Let $I$ be the index set, let $w_0$ be the longest Weyl-group element, and let $\Theta_{\mathbf i}^-(c)$, $\pi^+$, $\eta$, $\alpha_i(t)$, and the generalized minors $\Delta_{u\Lambda,v\Lambda}$ and characters $\chi_i$ be as in the construction. For a sign $p=(p_{i,j})_{i\ne j}$, write $\mathcal Y(p)$ for the associated monomial realization, and let $m_b(c)\in\mathcal Y(p)$ denote the monomial corresponding to a crystal element $b$.

Demazure-crystal expansion conjecture. There exists a reduced expression of the longest Weyl-group element $\mathbf i=(i_1,\ldots,i_N)$ and a sign $p=(p_{i,j})_{i\ne j}$ such that, for every $i\in I$, there are Demazure crystals $B^-_w(i)\subset B(\Lambda_k)$ and $B^+_{w'}(i)\subset B(\Lambda_j)$, together with positive integers $a_b$ and $a_{b'}$, satisfying

$$\chi_i(\pi^+(w_0^{-1}t\Theta_{\mathbf i}^-(c)))=\Delta_{w_0\Lambda_i,s_i\Lambda_i}(\Theta_{\mathbf i}^-(c))=\sum_{b\in B^-_w(i)}a_bm_b(c),$$ $$\chi_i(\pi^+(w_0^{-1}\eta(t\Theta_{\mathbf i}^-(c))))=\alpha_i(t)\Delta_{w_0s_i\Lambda_i,\Lambda_i}(\Theta_{\mathbf i}^-(c))=\alpha_i(t)\sum_{b'\in B^+_{w'}(i)}a_{b'}m_{b'}(c).$$

This conjecture proposes a uniform description of the relevant generalized minors by monomials indexed by Demazure crystals. The preceding type $A_n$ calculations provide explicit supporting examples, while the general existence of the reduced word, sign, and Demazure-crystal expansions remains unresolved in the supplied source.