Conjecture IV_FF on rational points for nonsplit fibrations

Let $f:X\to C$ be a fibration over a function field with $K$-core factorization $f=g_f\circ c_f$, and suppose the orbifold fibration

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

is not split. Conjecture IV_{FF}. There should exist a Zariski closed subset $T\subset C(f)$ with $T\neq C(f)$ such that, for every finite extension $K'/K$, the image $c_f(S(X'))$ is contained in $T$ together with a finite union of images of sections of $g_f$. The source gives no resolution.