The weight-vector conjecture for toric degenerations of flag varieties

Let $n\geq 3$ be an arbitrary integer, and let $\underline w_0$ be a reduced expression. For each Plücker variable, let ${\bf m}_J$ be the corresponding minimal monomial and let $e({\bf m}_J)$ denote its weight; write

$${\bf w}_{\underline w_0}=(e({\bf m}_1),e({\bf m}_2),\ldots,e({\bf m}_{2,3,\ldots,n}))$$

for the resulting weight vector. Let $\operatorname{trop}(\operatorname{\mathcal{F}\hspace{-1.6pt}\ell}_n)$ be the tropicalization of the flag variety, let MP denote the stated property of the string cone, and let $C$ be the corresponding prime cone with associated polytope. Weight-vector conjecture. For every reduced expression $\underline w_0$, the weight vector ${\bf w}_{\underline w_0}$ lies in the relative interior of a maximal cone in $\operatorname{trop}(\operatorname{\mathcal{F}\hspace{-1.6pt}\ell}_n)$. In particular, if the string cone satisfies MP, this vector lies in the relative interior of the prime cone $C$, whose associated polytope is combinatorially equivalent to the string polytope $Q_{\underline w_0}(\rho)$.