The generalized valuation conjecture for special sections and integral critical points

Let $G$ be a complex reductive group, let $B$ be a Borel subgroup, let $w_0$ be the longest Weyl-group element, and let $\mathbf{i}$ be a reduced expression of $w_0$. For a dominant weight $\lambda$, let $p_\lambda\in Z_{t^\lambda}(\mathcal{K}_{>0})^+$ be the critical point, and suppose that $p_\lambda$ is integral, meaning that $p_\lambda\in Z(\mathbb{C}((t)))$. Let $\omega_\lambda^{-1}\in H^0(G/B,\mathcal{L}_\lambda)$ be the corresponding special section, and let $\nu_{\lambda,\mathbf{i}}$ and $\nu^{\vee}_{\lambda,\mathbf{i}}$ be the maps to the string polytope associated with $\mathbf{i}$. The generalized valuation conjecture. Given $\lambda$ such that $p_\lambda$ is integral, we have

$$\nu_{\lambda,\mathbf{i}}(\omega_\lambda^{-1})=\nu^{\vee}_{\lambda,\mathbf{i}}(p_\lambda)$$

for all reduced expressions $\mathbf{i}$ of $w_0$. This generalizes the special case $\lambda=2\rho$ by relating the Newton–Okounkov valuation of each constructed special section to the tropical valuation of its integral critical point. The source gives no evidence that the conjecture has been proved or disproved.