In Section 7 of Alice Silverberg's Rank "Cheat Sheet", Silverberg stated

The Bhargava Conjecture: For each $n > > 1$ the average size of $S_{n}(E/\mathbb{Q})$ is $\displaystyle\sum\limits_{d|n} d$.

Here $S_{n}(E/\mathbb{Q})$ is the $n$-Selmer group of $E/\mathbb{Q}$. Silverberg remarks that assuming the Bhargava conjecture for infinitely many $n$, the parity conjecture, and the equidistribution of root numbers of $L$-functions of elliptic curves over $\mathbb{Q}$, it follows that $50$% of elliptic curves have rank $0$ and $50$% of elliptic curves have rank $1$ (this is called the Rank Distribution Conjecture).

Does anyone have a conjectural strategy for proving the equidistribution of root numbers?

Silverberg mentions the Poonen-Rains conjecture together with the parity conjecture implying the Rank Distribution Conjecture (which in turn implies equidistribution of root number), so it could be that trying to prove the Poonen-Rains conjecture offers a possible approach, but it seems to me that one in fact needs the equidistribution conjecture as a hypothesis in the latter statement of Silverberg...