Monomial realization conjecture for generalized minors and Demazure crystals

Let $I$ be the index set of the root system, let $\mathbf i=(i_1,\ldots,i_N)$ be a reduced expression of the longest Weyl-group element, and let $p=(p_{i,j})_{i\ne j}$ be the associated integer data. For each $i\in I$, let $\Theta_{\mathbf i}^-(c)$, $\pi^+$, $\eta$, $\chi_i$, and the generalized minors $\Delta_{w_0\Lambda_i,s_i\Lambda_i}$ and $\Delta_{w_0s_i\Lambda_i,\Lambda_i}$ be as in the preceding construction. Let $B^-_w(i)\subset B(\Lambda_k)$ and $B^+_{w'}(i)\subset B(\Lambda_j)$ be Demazure crystals, and let $m_b(c)\in\mathcal Y(p)$ denote the monomial corresponding to $b$ in the monomial realization associated with $p$. Monomial realization conjecture. There exists a reduced expression $\mathbf i$ of the longest Weyl-group element and data $p$ such that, for every $i\in I$, there are Demazure crystals $B^-_w(i)$ and $B^+_{w'}(i)$ and positive integers $a_b$ and $a_{b'}$ satisfying

$$\chi_i\bigl(\pi^+(w_0^{-1}t\Theta_{\mathbf i}^-(c))\bigr)=\Delta_{w_0\Lambda_i,s_i\Lambda_i}\bigl(\Theta_{\mathbf i}^-(c)\bigr)=\sum_{b\in B^-_w(i)}a_bm_b(c),$$

and

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

The conjecture proposes a simultaneous Demazure-crystal monomial expansion for the two generalized minors arising from the decorated geometric-crystal construction. It extends the explicitly verified type $A_n$ examples and would connect generalized minors, geometric crystals, and monomial realizations of crystal bases; the source provides no resolution.