The classification conjecture for smooth fibre-like polytopes in odd prime dimension

Let $\Delta$ be a smooth fibre-like polytope of dimension $d$, and let $X(\Delta)$ denote the associated smooth toric Fano variety. Assume that $d$ is an odd prime number. Classification conjecture. Then either

$$X(\Delta)=\mathbb{P}^d$$

or

$$X(\Delta)=(\mathbb{P}^1)^d.$$

This speculation proposes a complete classification in odd prime dimensions for the smooth fibre-like polytopes considered in the paper. The stated computations verify the analogous classification for smooth toric Fano varieties of dimension at most $8$, but the conjecture remains open in general.