The sRC quotient structure conjecture for klt pairs

Let $(X,\Delta)$ be a klt pair with $X$ smooth and $-(K_X+\Delta)$ nef. An orbifold morphism is a morphism of pairs, and a pair is slope rationally connected when its general orbifold fibers have the corresponding slope rational connectedness property. Then there is an orbifold morphism

$$\rho:(X,\Delta)\longrightarrow(R,\Delta_R)$$

such that $(R,\Delta_R)$ is a klt pair with $R$ smooth and $c_1(K_R+\Delta_R)=0$, the general orbifold fibers $(X_r,\Delta_r)$ are slope rationally connected, and the fibration is locally trivial with respect to pairs: for every sufficiently small open set $U\subset R$,

$$(\rho^{-1}(U),\Delta)\cong(U,\Delta_R|_U)\times(X_r,\Delta_{X_r}),$$

where $X_r$ is a general fiber of $\rho$.

sRC quotient structure conjecture. Such an orbifold morphism with all the stated properties exists.

The conjecture concerns a structure theorem for slope rationally connected quotients, which generalize MRC fibrations while incorporating the boundary. It was solved for log smooth surfaces, but remains open in general.