Monical–Tokcan–Yong's Bruhat-order polytope conjecture

Let $\alpha,\beta\in\mathbb{Z}^n$, and let $\leq_{S}$ be the Bruhat order on $\mathbb{Z}^n$ generated by the relations $\lambda>s_{ij}(\lambda)$ for $1\leq i<j\leq n$ and $\lambda_i<\lambda_j$, together with $s_{ij}(\lambda)>_{S}\lambda+e_i-e_j$ when $\lambda_j-\lambda_i>1$. Define

$$\widehat{\mathcal{P}}_{\alpha}=\operatorname{conv}\{\gamma\in\mathbb{Z}^n:\gamma\leq_{S}\alpha\}.$$

Monical–Tokcan–Yong's conjecture. If $\beta\in\widehat{\mathcal{P}}_{\alpha}$ and $\beta\in\mathbb{Z}^n_{\geq 0}$, then $\beta\leq_{S}\alpha$.

The paper states that its Theorem resolves this conjecture in the affirmative, so the claim is treated as solved.