# Primitive collections as lattice generators for the Mori cone

I am looking at the following theorem from "Toric Varieties" by Cox, Little and Schenk:

Theorem 6.4.11: If $$X_{\Sigma}$$ is a simplicial toric variety, then $$\overline{NE}(X_{\Sigma}) = \large{\Sigma}_P \mathbf{R}_{\geq 0}r(P)$$, where the sum is over all primitive collections $$P$$.

Such primitive collections $$P$$ gives rise to a 1-cycle in $$N_1(X_{\Sigma})$$ represented by an element $$r(P) = (b_{\rho})_{\rho \in \Sigma(1)} \in \mathbf{R}^{\Sigma(1)}$$, which can be shown to represent an effective class. Assuming that $$X_{\Sigma}$$ is a smooth projective variety, it is easy to show that the numbers $$b_{\rho}$$ are integral, in which case $$r(P)$$ sits inside $$\overline{NE}(X_{\Sigma}) \cap \mathbf{Z}^n$$.

Apparently, it is not known whether the $$r(P)$$'s always generate $$\overline{NE}(X_{\Sigma}) \cap \mathbf{Z}^n$$ as a semigroup.

However, I'm interested in the low-dimensional cases, specifically dimension 3.

My question is: Is it known, or can it be shown that the $$r(P)$$'s generate $$\overline{NE}(X_{\Sigma}) \cap \mathbf{Z}^n$$ for $$X_{\Sigma}$$ smooth, projective and of dimension $$3$$?