Determinant conjecture for orbit equivalence relations

Suppose $\Gamma\overset{\alpha}{\curvearrowright}(X,\mu)$ is a probability-measure-preserving action, with associated invariant random subgroup $H_\alpha$ and orbit equivalence relation $\mathcal{R}_\alpha$. The IRS $H_\alpha$ satisfies the determinant conjecture when its associated trace obeys the modified Fuglede–Kadison determinant inequality for every matrix over $\mathbb{Z}[\Gamma]$. Determinant conjecture for orbit equivalence relations. If $H_\alpha$ satisfies the determinant conjecture as an IRS, then the orbit equivalence relation $\mathcal{R}_\alpha$ also satisfies Lück's determinant conjecture. This would transfer the determinant property from an invariant random subgroup to its associated measured orbit equivalence relation, extending the known implication from co-sofic IRSs and addressing the possible obstruction from the intermediate $L^\infty(X)$ algebra.