Bilu's conjecture on unbounded $n$-rank of class groups

Let $n>1$ be an integer, let $F$ be a number field, and let $d$ be a positive integer. For a number field $L$, write $Cl(L)[n]$ for the subgroup of the ideal class group of $L$ annihilated by $n$, and let $\operatorname{rk} Cl(L)[n]$ denote its rank. Bilu's conjecture. The quantity

$$\operatorname{rk} Cl(L)[n]$$

is unbounded as $L$ varies over the extensions of $F$ of degree $d$. This conjecture predicts unbounded torsion rank in class groups within every fixed-degree extension family; the source presents the paper's elliptic-curve constructions as modest evidence toward it, without indicating a resolution.