The first Zassenhaus conjecture for normalized units of integral group rings

Let $G$ be a finite group. Write $\mathbb{Z}G$ for its integral group ring and

$$\operatorname{V}(\mathbb{Z}G)=\{u\in \mathcal{U}(\mathbb{Z}G):\varepsilon(u)=1\}$$

for the group of normalized units, where $\varepsilon$ is the augmentation map. First Zassenhaus conjecture. For any $u \in \operatorname{V}(\mathbb{Z}G)$ of finite order, there exist an element $g \in G$ and a unit $x$ in the rational group algebra $\mathbb{Q}G$ such that

$$x^{-1}ux=g.$$

This conjecture concerns whether every torsion unit of augmentation one in an integral group ring is rationally conjugate to a group element. It has been proved for several classes of finite groups, including nilpotent groups, groups with a normal Sylow subgroup and abelian complement, and cyclic-by-abelian groups.