Classification conjecture for toric 2-Fano manifolds

A toric 2-Fano manifold is a smooth projective toric variety that is Fano and whose second Chern character satisfies

$$\operatorname{ch}_2(X)\cdot S>0$$

for every surface $S\subset X$. Projective spaces are examples of toric 2-Fano manifolds.

Toric 2-Fano classification conjecture. The only toric 2-Fano manifolds are projective spaces.

The conjecture would complete the classification of toric 2-Fano manifolds. Despite extensive work on special classes, projective spaces remain the only known examples.