The rank-zero fibre conjecture for non-isotrivial elliptic surfaces

Let $\pi:X\to \mathbb{P}^1$ be a non-isotrivial elliptic surface defined over $\mathbb{Q}$, and let $\mathrm{MW}(X,\pi)$ be its group of sections defined over $\mathbb{Q}$. Assume

$$\operatorname{rk}\mathrm{MW}(X,\pi)=0.$$

Rank-zero fibre conjecture. There are infinitely many $b\in\mathbb{P}^1(\mathbb{Q})$ such that

$$\operatorname{rk}X_b(\mathbb{Q})=0;$$

that is, $\mathscr{N}(X,\pi)$ is infinite.

This is a special case of the preceding conjecture and remains open; the paper states that no example of a non-isotrivial elliptic surface with infinite $\mathscr{N}(X,\pi)$ is known.