Let $X$ be a complex Fano manifold such that each extremal ray of $\overline{\text{NE}(X)}_{\mathbb{R}}$ is generated by a primitive class in $H_2(X;\mathbb{Z})$ of a free rational curve. Thus, the extremal rays are all fiber type. (The "primitive" hypothesis rules out, e.g., conic bundles where "half" of the fiber class is integral.)

**Question.** Is the cone of effective curves in $H_2(X;\mathbb{Z})$ a free $\mathbb{Z}_{\geq 0}$-semigroup, $\mathbb{Z}_{\geq 0}^r$?

There are many positive examples, e.g., Fano manifolds with a transitive (algebraic) action of a complex Lie group and complete intersections in these of low degree. However, for the general question, the best results that I have found are in the following article of Jaroslaw Wisniewski.

MR1639552 (2000e:14018)

Wiśniewski, Jarosław A.

Cohomological invariants of complex manifolds coming from extremal rays.

Asian J. Math. 2 (1998), no. 2, 289–301.

https://arxiv.org/abs/math/9803010