Conjecture on smooth slope rationally connected quotients for nef logarithmic tangent bundles

Let $(X,D)$ be a log smooth pair, meaning that $X$ is smooth and $D$ has simple normal-crossing support. Its logarithmic tangent bundle is denoted by $T_X(-\log D)$. An sRC-quotient is an orbifold morphism whose general fiber is slope rationally connected and whose base pair has pseudo-effective log canonical divisor. Smooth sRC-quotient conjecture. If $T_X(-\log D)$ is nef, then one can take an sRC-quotient that is a smooth morphism.

This conjecture proposes smoothness of the slope rationally connected quotient under a positivity condition on the logarithmic tangent bundle. The source gives no resolution in general; the preceding quotient conjecture is known for smooth surfaces.