Bogomolov's conjecture for non-isotrivial fibrations

Let $X$ be a smooth projective surface over a field $k$, let $Y$ be a smooth projective curve over $k$, and let $f:X\to Y$ be a semistable fibration whose smooth generic fiber $C/K$ has genus $\bar{g}\geq 2$. Let $K$ be the function field of $Y$, with algebraic closure $\overline{K}$, and let $r_C(P)$ be defined from the Néron–Tate seminorm on $\operatorname{Pic}^0(C)(\overline{K})$ as in the setup. Bogomolov conjecture. If $f$ is non-isotrivial, then $r_C(P)>0$ for all $P\in\operatorname{Pic}^0(C)(\overline{K})$. This asserts a positive lower bound for the radius beyond which the corresponding bounded-height sets on the curve become infinite; the source gives no resolution status.