Homology-gradient approximation by Hughes-free division rings

Let $\mathbb{F}$ be a skew field. Let $G$ be a torsion-free residually finite group of type $\mathsf{FP}_{n+1}(\mathbb F)$ such that the Hughes-free division ring $\mathcal D_{\mathbb{F}G}$ exists. Let $(G_i)_{i\in\mathbb{N}}$ be a residual chain of finite-index normal subgroups. The approximation conjecture. For all $m\leqslant n$,

$$b^{(2)}_{m}(G,(G_i);\mathbb{F})=b^{\mathcal D_{\mathbb{F}G}}_m(G).$$

In particular, the defining limit supremum is a genuine limit and is independent of the residual chain. The conjecture is motivated by the agreement, over $\mathbb{Q}$ and for RFRS groups, between the Hughes-free division ring and the Linnell skew field, whose Betti numbers agree with the $\ell^2$-Betti numbers; no resolution is given here.