Conjecture III_FF on potential density over function fields

Let $f:X\to C$ be a fibration over a function field, with $K$-core factorization $f=g_f\circ c_f$ and base orbifold $(C(f)/\Delta(c(f)))$. Conjecture III_{FF}. The $K'$-rational points of $X_K$ are Zariski dense in $X$ for some finite extension $K'/K$ if and only if

$$g_f:(C(f)/\Delta(c(f)))\to C$$

is split. In particular, if $X_K$ is special, then $X(K')$ is Zariski dense in $X$ for some finite extension $K'/K$. The source gives no resolution.