By Remark 3.21, the $\nabla u$ introduced in the above proof is actually the gradient of the extension $u$ over $N_{\mathbb{R}}$. By the proof above, we know $\langle\nabla u,w_{i}\rangle=1$ on $\text{Star}(w_{i})\subset\partial\Delta_{\lambda}^{\vee}$. This directional derivative can only increase as $x\in N_{\mathbb{R}}$ moves in the $w_{1}$-direction. But $\nabla u\in\Delta$ on $N_{\mathbb{R}}$ since the extension is admissible, so $\langle\nabla u,w_{i}\rangle\leq 1$ everywhere, hence the claim. ∎