The base-change conjecture for division closures of group algebras

Let $L/K$ be an extension of subfields of $\mathbb{C}$, and let $G$ be a group satisfying the strong Atiyah conjecture. Let $\mathcal D_{K[G]}$ and $\mathcal D_{L[G]}$ be, respectively, the division closures of $K[G]$ and $L[G]$ in $\mathcal U(G)$. Base-change conjecture for division closures. The ring $\mathcal D_{L[G]}$ is isomorphic to the classical ring of quotients of

$$\mathcal D_{K[G]}\otimes_K L.$$

This property is established in the paper for locally indicable groups and is known for certain sofic groups satisfying the strong Atiyah conjecture. The source explicitly presents the assertion as an expectation in general, so it remains open.