The only toric 2-Fano manifold with m(X)=2 is the projective plane P^2.
Birational Geometry of Toric Varieties , NOTE =
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On the classification of toric $2$-Fano manifolds: generic $\mathbb{P}^2$-bundles
The only toric 2-Fano manifold with m(X)=2 is the projective plane P^2.