The class T Atiyah and universal-localization conjecture

Let $G$ be a group in the class $\mathcal{T}$ with finite $\operatorname{lcm}(G)$, and let $K\subseteq\mathbb{C}$ be any subfield closed under complex conjugation. Let $\Sigma(K,G)$ be the set of matrices over $K[G]$ that become invertible in the relevant localization, and let $\mathcal{R}_{K[G]}$ denote the $*$-regular closure. Class $\mathcal{T}$ conjecture. The group $G$ satisfies the strong, algebraic and center-valued Atiyah conjectures over $K$, and $\mathcal{R}_{K[G]}$ is the universal localization of $K[G]$ over $\Sigma(K,G)$. The paper presents this as a further conjecture extending results known for Linnell's class and certain locally indicable groups; its general validity remains open.