Zhang's conjecture on $l$-torsion in class groups

Let $K$ be a number field, let $n=[K:\mathbb{Q}]$, and let $\epsilon>0$. Write $Cl(K)[l]$ for the subgroup of elements of the class group $Cl(K)$ annihilated by $l$, and let $D_K$ denote the discriminant of $K$. Zhang's conjecture. For any number field $K$ and fixed $n$ and $l$, one has

$$|Cl(K)[l]|\ll_{\epsilon,n,l}D_K^{\epsilon}.$$

This is a strong asymptotic conjecture asserting that the $l$-torsion in class groups grows more slowly than every positive power of the discriminant. The source gives no resolution status for the conjecture.