Smoothness-frequency conjecture for abelian varieties

Let $k$ be a number field, let $A$ be an abelian variety over $k$, and let $S_A$ be the set of places of good reduction for $A$. For $u\in(0,1)$, call an integer $B^u$-smooth if all its prime factors are at most $B^u$. Smoothness-frequency conjecture. For every $u\in(0,1)$,

$$\limsup_{B\to\infty}\frac{\#\{v\in S_A:\#\mathbb F_v\le B\text{ and }\#A(\mathbb F_v)\text{ is }B^u\text{-smooth}\}}{\#\{v\in S_A:\#\mathbb F_v\le B\}}>0.$$

This conjecture formalizes the smoothness assumption used in the paper's random-model evidence for the finite-place conjecture. It is presented as a theoretical heuristic assumption and is not proved in the supplied text.