The Poonen–Rains rank conjecture for elliptic curves

Let $F$ be a number field and order elliptic curves over $F$ by height.

Poonen–Rains rank conjecture. For each $r\in\{0,1\}$, $50\%$ of elliptic curves over $F$ have Mordell–Weil rank $r$.

This is presented as a consequence of the Poonen–Rains Selmer-distribution conjecture together with the Birch and Swinnerton-Dyer conjecture. The source gives no resolution status.