Lück's determinant conjecture for all discrete groups

Let $\Gamma$ be a discrete group, let $L(\Gamma)$ be its group von Neumann algebra, and let $\tau$ be the standard trace on $L(\Gamma)$. For $A\in M_n(\mathbb{Z}[\Gamma])$, write

$$\ln_+(t)=\begin{cases}0 & t=0,\\ \ln(t) & t>0,\end{cases}$$

and let $\operatorname{tr}_n$ denote the normalized trace on $M_n(\mathbb{C})$. The group $\Gamma$ satisfies Lück's determinant conjecture when

$$(\tau\otimes\operatorname{tr}_n)\bigl(\ln_+(A^*A)\bigr)\geq 0$$

for every $A\in M_n(\mathbb{Z}[\Gamma])$ and every $n\in\mathbb{N}$. Lück's determinant conjecture. Every discrete group satisfies Lück's determinant conjecture. The quantity is the logarithm of the modified Fuglede–Kadison determinant, obtained by ignoring zero singular values. The conjecture asks whether this determinant inequality holds universally for groups; the source presents it as an open possibility and motivates its extension to invariant random subgroups, where determinant-conjecture examples exist that are not co-sofic.