Mazur's conjecture on ranks in elliptic K3 surface families

Let $\pi:\mathcal{E}\to\mathbf{P}^1_{\mathbf{Q}}$ be an elliptic K3 surface, and for $t\in\mathbf{P}^1_{\mathbf{Q}}$ let $\mathcal{E}_t=\pi^{-1}(t)$ be the fibre. Mazur's conjecture. The family $\{\mathcal{E}_t\}_{t\in\mathbf{P}^1_{\mathbf{Q}}}$ satisfies exactly one of the following conditions: either the elliptic curve $\mathcal{E}_t$ has Mordell–Weil rank $0$ for all but finitely many $t\in\mathbf{P}^1(\mathbf{Q})$, or

$$\{t\in\mathbf{P}^1(\mathbf{Q}):\operatorname{rank}(\mathcal{E}_t(\mathbf{Q}))>0\}$$

is dense in $\mathbf{P}^1(\mathbf{R})$. This is presented as a consequence of Mazur's conjecture for elliptic K3 surfaces; the source does not give evidence that it has been resolved.