Campana–Cao–Matsumura's slope rationally connected quotient conjecture

Let $X$ be a smooth projective variety and let $D$ be an effective $cmathbb{Q}$-divisor on $X$. Assume that $(X,D)$ is klt and that $-(K_X+D)$ is nef. An orbifold morphism is a morphism of orbifold pairs; write it as

$$\rho:(X,D)\longrightarrow(R,D_R).$$

Campana–Cao–Matsumura's conjecture. There exists an orbifold morphism $\rho:(X,D)\to(R,D_R)$ such that:

1. $(R,D_R)$ is a klt pair and $c_1(K_R+D_R)=0$.
2. For a general point $r\in R$, the general fiber $(X_r,D_r)$ is slope rationally connected.
3. $\rho$ is locally trivial with respect to pairs.

This conjecture predicts a structural quotient for klt pairs with nef anticanonical divisor. It is known for smooth surfaces, while the general statement remains open.