K-polystable toric Fano varieties with small alpha invariants

For every $n\ge 2$, we exhibit an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety $X_n$, defined by the face fan of an explicit lattice polytope, and whose alpha invariant is exactly $\frac{2}{2n+1}$. This answers a question of Liu and Zhuang whether there exists an $n$-dimensional K-semistable $\mathbb{Q}$-Fano variety whose alpha invariant is between $\frac{1}{n+1}$ and $\frac{1}{n}$. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.