The integer-operator conjecture for group-measure-space rings

Let $(X,\mu)$ be a standard probability space, and let $\Gamma$ be a discrete group acting on $(X,\mu)$ by measure-preserving Borel isomorphisms. A normal operator is integer when its spectral measure is an integer measure on $\mathbb C$. Consider the algebraic crossed-product subring

$$L^{\infty}(X,\mathbb Z)\rtimes_{\rm alg}\Gamma\subset L^{\infty}(X,\mathbb C)\rtimes\Gamma.$$

The integer-operator conjecture for group-measure-space rings. Every normal operator in this subring is an integer operator. This is the group-measure-space analogue of the conjecture for normal elements of $\mathbb Z\Gamma$; the source describes the evidence for it as less striking.