Measure-theoretic determinant conjecture

Let ${\mathcal R}$ be a standard equivalence relation on a standard probability space. Its groupoid ring is ${\mathbb Z}{\mathcal R}$, and ${\mathcal N}({\mathcal R})$ is the associated finite von Neumann algebra. The generalized Fuglede–Kadison determinant of a matrix $A\in M(m\times n,{ \mathbb Z}{\mathcal R})$ is denoted by ${\operatorname{det}}_{{\mathcal N}({\mathcal R})}(A)$. Measure-theoretic determinant conjecture. Every standard equivalence relation satisfies

$${\operatorname{det}}_{{\mathcal N}({\mathcal R})}(A)\geq 1$$

for every such matrix $A$. This conjecture extends the determinant conjecture from groups to measured equivalence relations. The paper establishes it in several cases, including Bernoulli actions of countable residually amenable groups, but the general assertion remains open.