On isomorphity of measure-preserving $\mathbb Z^2$-actions that have isomorphic Cartesian powers

Assume that $\Delta$ and $\Pi$ are representations of the group $\mathbb Z^2$ by operators on the space $L_2(X,\mu)$ that are induced by measure-preserving automorphisms, and for some $d$, the representations $\Delta^{\otimes d}$ and $\Pi^{\otimes d}$ are conjugate to each other, $\Delta\bigl(\mathbb Z^2\setminus(0,0)\bigr)$ consists of weakly mixing operators, and there is a weak limit (over some subsequence in $\mathbb Z^2$ of operators from $\Delta(\mathbb Z^2)$) which is equal to a nontrivial, convex linear combination of elements of $\Delta(\mathbb Z^2)$ and of the projection onto constant functions. We prove that in this case, $\Delta$ and $\Pi$ are also conjugate to each other.