Generalized measurable Sarnak's Conjecture

Let $\boldsymbol{K}=(K,\mathcal{O}_{K},D,\mathcal{B}=\{b_{1},\dots,b_{D}\})$ be an integral tuple and $\chi\in\mathcal{M}^{a}_{K}$. Let $(X,\mu,T_{1},\dots,T_{d})$ be a measure-preserving system with commuting transformations $T_{1},\dots,T_{d}$, and let $\Phi\in L^{\infty}(\mu)$. Generalized measurable Sarnak's conjecture. For $\mu$-almost every $x\in X$,

$$\lim_{N\to\infty}\frac{1}{N^{D}}\sum_{1\leq n_{1},\dots,n_{D}\leq N}\chi(n_{1}b_{1}+\dots+n_{D}b_{D})\Phi(T_{1}^{n_{1}}\cdot\ldots\cdot T_{D}^{n_{D}}x)=0.$$

Unlike the topological formulation, this conjecture makes no assumption on entropy. It asks for Möbius-type disjointness in the measure-theoretic setting over an arbitrary number field; the source gives no resolution.