The conjecture that the first $L^2$-Betti numbers agree over the von Neumann algebra and division ring

Let $G$ be a finitely presented group and let $F$ be a field. Denote by $b^{(2)}_1(G;\mathcal N(G))$ the first $L^2$-Betti number with coefficients in the group von Neumann algebra, and by $b^{(2)}_1(G;\mathcal D_{FG})$ the corresponding invariant with coefficients in the division ring $\mathcal D_{FG}$. First $L^2$-Betti-number equality conjecture. The equality

$$b^{(2)}_1(G;\mathcal N(G))=b^{(2)}_1(G;\mathcal D_{FG})$$

holds for all finitely presented groups $G$ and every field $F$.

This conjecture compares characteristic-independent von Neumann-algebra invariants with division-ring versions over arbitrary fields. No resolution is given in the supplied source.