Mazur's conjecture on the topology of positive-rank fibers

Let ${\mathcal E} \to {\mathbb P}^1_{\mathbb Q}$ be an elliptic surface with base ${\mathbb P}^1_{\mathbb Q}$. For $t \in {\mathbb P}^1(\mathbb Q)$, consider the fiber ${\mathcal E}_t$ when it is an elliptic curve, and its Mordell–Weil rank.

Mazur's conjecture. One of the following two conditions holds:

1. For all but finitely many $t \in {\mathbb P}^1(\mathbb Q)$, the fiber ${\mathcal E}_t$ is an elliptic curve with Mordell–Weil rank equal to zero.
2. The set of $t \in {\mathbb P}^1(\mathbb Q)$ such that ${\mathcal E}_t$ is an elliptic curve with positive Mordell–Weil rank is dense in ${\mathbb P}^1(\mathbb R)$.

This is one of Mazur's conjectures on the topology of rational points on varieties. The source gives no resolution status, so the conjecture is recorded as open.