The Kleiman–Mori cone conjecture for complete toric varieties

Let $X$ be a non-singular complete toric variety.

Kleiman–Mori cone conjecture.

$$\overline{NE}(X)=NE(X)\subsetneq N_1(X).$$

This conjecture proposes that the Kleiman–Mori cone of a complete toric variety is closed but is a proper subset of the space of numerical curve classes. It concerns the possibility of complete toric varieties whose cone of effective curves does not fill $N_1(X)$.