Orbit equivalence rigidity for product actions

Let ${\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_n$ be hyperbolic, property (T) groups, for some $n\ge 1$. We prove that if a product $\Gamma_1\times\dots\times\Gamma_n \curvearrowright X_1\times\dots\times X_n$ of measure preserving actions is stably orbit equivalent to a measure preserving action $\Lambda\curvearrowright Y$,then$\Lambda\curvearrowright Y$isinducedfromanaction$\Lambda_0\curvearrowright Y_0$ such that there exists a direct product decomposition ${\mathrm{\Lambda }}_0={\mathrm{\Lambda }}_1\times \cdots \times {\mathrm{\Lambda }}_n$ into $n$ infinite groups. Moreover, there exists a measure preserving action ${\mathrm{\Lambda }}_i\curvearrowright Y_i$ that is stably orbit equivalent to ${\mathrm{\Gamma }}_i\curvearrowright X_i$, for any $1\le i\le n$, and the product action ${\mathrm{\Lambda }}_1\times \cdots \times {\mathrm{\Lambda }}_n\curvearrowright Y_1\times \cdots \times Y_n$ is isomorphic to ${\mathrm{\Lambda }}_0\curvearrowright Y_0$.