Averages of root numbers in elliptic-surface families

Let $\pi\colon X\to\mathbb{P}^1$ be an elliptic surface defined over $\mathbb{Q}$. Fix an affine chart $\mathbb{A}^1\subseteq\mathbb{P}^1$ with affine coordinate $t$, and suppose that $\pi$ has a fibre of multiplicative reduction on $\mathbb{A}^1$. For smooth fibres at integer parameters, consider their global root numbers $w(X_n)$. Then

$$\lim_{X\to\infty}\frac{1}{X}\sum_{\substack{1\le n\le X\\ \pi\text{ has a smooth fibre at }t=n}}w(X_n)=0.$$

Averages of root numbers conjecture. The average root number in this family is zero. Helfgott gives strong support and proves the assertion for several elliptic surfaces under suitable analytic-number-theory hypotheses, but it is not known in the stated generality.