Positivity of the rank for elliptic surface fibres

Let $\pi\colon X\to\mathbb{P}^1$ be an elliptic surface defined over $\mathbb{Q}$, and fix an affine chart $\mathbb{A}^1\subseteq\mathbb{P}^1$. Suppose that the elliptic surface has a fibre of multiplicative type over some point of $\mathbb{A}^1$. Define

$$N(x)=\#\{n\in\mathbb{Z}:1\le n\le x\text{ and the fibre }X_n\text{ is an elliptic curve with }\operatorname{rank}X_n(\mathbb{Q})>0\}.$$

Positivity of the rank. There is $c>0$ such that for all sufficiently large $x$, $N(x)\ge c\cdot x$. This conjecture is motivated by work of Helfgott and, together with a parity statement for elliptic curves, is used to obtain positive Diophantine definability results.