  One of the central aims of the Minimal Model Program is to show that a
projective log canonical pair $(X,\Delta)$ with $K_X+\Delta$ pseudoeffective
has a good model, i.e.\ a minimal model $(Y,\Delta_Y)$ such that $K_Y+\Delta_Y$
is semiample. The goal of this paper is to show that this holds if $X$ is
uniruled but not rationally connected, assuming the Minimal Model Program in
dimension $\dim X-1$. Moreover, if $X$ is rationally connected, then we show
that the existence of a good minimal model for $(X,\Delta)$ follows from a
nonexistence conjecture for a very specific class of rationally connected pairs
of Calabi--Yau type.
