Roughly speaking, this question asks whether there is a measure-conjugacy between two transformations if there are measure-conjugacies between their ergodic components.

Suppose $(X,\mu)$ is a standard probability space and $T$ and $S$ are measure-preserving transformations of $(X,\mu)$. By the ergodic decomposition theorem, there are standard probability spaces $(Y,\nu), (Z,\zeta)$ and Borel maps $\phi:Y \to M_1(X), \psi:Z\to M_1(X)$ (where $M_1(X)$ is the space of Borel probability measures on $X$) such that for a.e. $y \in Y$, $\phi(y)$ is ergodic, $T$-invariant and $\int \phi(y)~d\nu(y)=\mu$. Similarly, for a.e. $z\in Z$, $\psi(z)$ is ergodic and $S$-invariant and $\int \psi(z)~d\zeta(z)=\mu$.

Now suppose there is a measure-space isomorphism $\Omega:(Y,\nu) \to (Z,\zeta)$ such that for a.e. $y \in Y$, $(T,X,\phi(y))$ is measurably conjugate to $(S,X,\psi(\Omega(y)))$. Then is $T$ measurably conjugate to $S$?