The rank distribution conjecture for quadratic twists ordered by canonical height

Let $E/\mathbb{Q}$ be the fixed elliptic curve given by $y^2=x^3+Ax+B$, and let $E_d$ be its quadratic twist for squarefree $d$. Let $\mathcal{H}(Y)$ denote the set of twist parameters being considered with canonical height at most $Y$, and write $\operatorname{rank} E_d(\mathbb{Q})$ for the algebraic Mordell–Weil rank.

Rank distribution conjecture. For $\iota\in\{1,2\}$, one has

$$\#\{d\in\mathcal{H}(Y):\operatorname{rank} E_d(\mathbb{Q})=\iota\}\sim\frac{1}{2}\#\mathcal{H}(Y).$$

The conjecture predicts that, among twists ordered by the canonical height of their lowest non-torsion rational point, half have rank one and half have rank two. It is motivated by the expected equidistribution of the functional-equation sign and the expectation that analytic ranks, and hence algebraic ranks under Birch–Swinnerton-Dyer, are as small as permitted by parity.