Structure conjecture for sRC quotients of smooth klt pairs

Let $(X,D)$ be a klt pair with $X$ smooth and $-(K_X+D)$ nef. An orbifold morphism is a morphism of orbifold pairs. The pair $(X,D)$ is sRC if its slope rationally connected quotient has the vanishing property described in the preceding definition. An orbifold fiber $(X_r,D_r)$ is a fiber of such a morphism equipped with its induced orbifold structure.

Structure conjecture for sRC quotients. There exists an orbifold morphism

$$\rho:(X,D)\rightarrow(R,D_R)$$

with the following properties:

1. $(R,D_R)$ is a klt pair, $R$ is smooth, and

$$c_1(K_R+D_R)=0.$$

2. General orbifold fibers $(X_r,D_r)$ are sRC.
3. The fibration is locally trivial with respect to pairs: for any small open set $U\subset R$,

$$(\rho^{-1}(U),D)\cong(U,D_R|_U)\times(X_r,D_{X_r})$$

over $U\subset Y$, where $X_r$ is a general fiber of $\rho$.

This conjecture seeks a structure theorem for sRC quotients analogous to the known structure theorem for the sRC quotient of an lc pair. It predicts that, under nefness of the anti-canonical divisor, the quotient has a smooth klt base with numerically trivial log canonical class, sRC general fibers, and local product structure; its resolution status is not specified in the source.