Conjecture III_SD on dominated fibrations and rational quotients

Let $f:X\to C$ be a fibration over a function field, let $r_K:X\to R(X_K)$ be its rational quotient, and call $X_K$ dominated when it admits the dominant product-type rational map defined in the source. Conjecture III_{SD}. The following are equivalent: (i) $X_K$ is dominated if and only if $R(X_K)$ is split; equivalently, in the weaker formulation, if $X_K$ is dominated but not split, then $X_K$ is uniruled. The source presents the weaker form as equivalent to the first and gives no resolution.