The rank-jump density conjecture for elliptic curves over rational function fields

Let $\mathcal{E}$ be an elliptic curve over the function field $\Q(T)$ with nonconstant $j$-invariant. For $t\in\Q$, let $\mathcal{E}_t$ denote its specialization when it is an elliptic curve, and let $\operatorname{rank}\mathcal{E}(\Q(T))$ and $\operatorname{rank}\mathcal{E}_t(\Q)$ denote the corresponding Mordell–Weil ranks. The rank-jump density conjecture. The set

$$\left\{t\in\Q:\operatorname{rank}\mathcal{E}_t(\Q)\geq 2+\operatorname{rank}\mathcal{E}(\Q(T))\right\}$$

has density zero. This predicts that specializations with a rank increase of at least two are rare. The source provides no resolution, so the conjecture remains open.