Average-rank conjecture for elliptic surface specializations

Let $\mathcal{E}$ be the elliptic surface, let $\mathcal{E}_t$ be its fiber, and write $r_t=\operatorname{rank}\mathcal{E}_t(\mathbb{Q})$. Let $w_{\mathcal{E}_t}$ be the root number and put $r=\operatorname{rank}\mathcal{E}(\mathbb{Q}(T))$. Define

$$\mathcal{T}=\left\{t\in\mathbb{Z}:w_{\mathcal{E}_t}=(-1)^{r+1}\right\},$$

where $\delta(\mathcal{T})$ is the natural density of $\mathcal{T}$ in $\mathbb{Z}$.

Average-rank conjecture. One has

$$\lim_{X\to\infty}\frac{1}{2X}\sum_{|t|\le X}r_t=r+\delta(\mathcal{T}).$$

This conjecture predicts the average rank of the fibers from the generic rank and the density of fibers with the indicated root-number parity. The source gives heuristic and experimental motivation but no resolution status.