Conjecture on degenerations of rationally connected varieties

Let $R$ be a DVR with fraction field $K$ and residue field $k$. Let $X$ be a flat, projective $R$-scheme such that the generic fiber $X_K$ is smooth and separably rationally connected. Degeneration conjecture. The special fiber $X_k$ admits a $k$-morphism from a smooth projective separably rationally connected $k$-variety. The paper states that this standard conjecture would imply the corresponding geometric $C_1$ result for untilts of $\overline{\mathbb{F}}_p((t^{1/p^{\infty}}))$; its general validity remains open.