Park–Poonen–Voight–Wood conjecture on the distribution of elliptic-curve ranks

Let elliptic curves over $\mathbb{Q}$ be ordered by height, and let algebraic rank denote Mordell–Weil rank. Park–Poonen–Voight–Wood conjecture. (i) The proportion of elliptic curves with algebraic rank $0$ and the proportion with algebraic rank $1$ are both $\frac 12$. (ii) There are only finitely many elliptic curves with algebraic rank $>21$. (iii) For $1\leq r\leq 20$, the proportion of elliptic curves over $\mathbb{Q}$ with algebraic rank at least $r$ and height at most $X$ is $X^{\frac{21-r}{24}+o(1)}$. This refines the minimalist conjecture by predicting the distribution of higher algebraic ranks; the source attributes it to Park, Poonen, Voight, and Wood and does not state a resolution.