On the $2$-torsion in class groups of number fields

In $2020$, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao proved that for a finite extension $K/\mathbb{Q}$ of degree $n\geq 5$, the size of the $2$-torsion class group is bounded by $\# h_{2}(K)=O_{n,\varepsilon}(D_{K}^{\frac{1}{2}-\frac{1}{2n}+\varepsilon})$, where $D_{K}$ is the absolute discriminant of $K$. In the present paper, we improve their bound by proving that $\# h_{2}(K)=O_{n,\varepsilon}(D_{K}^{\frac{1}{2}-\frac{1}{2n}-\delta_{K}+\varepsilon})$, for a constant $\delta_{K}\geq\frac{1}{28n}-\frac{3}{28n(n-1)}$.