Conjecture on the average size of the BSD factor for elliptic curves

Let $\varepsilon>0$ be fixed. Let $\mathcal{F}_{\geq 1}(X)$ be the family of elliptic curves under consideration with positive rank, let $H(E_{A,B})$ be their height, and let $\mu(E_{A,B})$ be the arithmetic factor appearing in the Birch–Swinnerton-Dyer formula.

BSD-factor growth conjecture.

$$\lim_{X\to\infty}\frac{\#\{E_{A,B}\in\mathcal{F}_{\geq 1}(X):\mu(E_{A,B})>H(E_{A,B})^{1/12-\varepsilon}\}}{\#\mathcal{F}_{\geq 1}(X)}=1.$$

The conjecture asserts that, for almost all curves in the positive-rank family, the BSD factor $\mu(E_{A,B})$ exceeds the indicated power of the height. It is presented as a heuristic consequence of predictions that the Tate–Shafarevich group is often small, and no resolution is supplied in the source.