Equidistribution of root numbers in elliptic-surface families

Let $\pi\colon X\to\mathbb{P}^1$ be an elliptic surface defined over $\mathbb{Q}$, choose an affine chart $\mathbb{A}^1\subseteq\mathbb{P}^1$ with coordinate $t$, and suppose that $\pi$ has a fibre of multiplicative reduction on $\mathbb{A}^1$. For $\epsilon\in\{-1,1\}$, define

$$S_\epsilon=\{n\in\mathbb{N}:\pi\text{ has a smooth fibre at }t=n\text{ and }w(X_n)=\epsilon\}\subseteq\mathbb{N}.$$

Equidistribution of root numbers conjecture. Both $S_{-1}$ and $S_1$ have natural density $1/2$ in $\mathbb{N}$. This is the equivalent sign-equidistribution formulation of the conjectured zero average of root numbers, and remains open in general.