Given a submodular function $f$ on $\Omega=X_1\cup X_2$ where $X_1$ and $X_2$ are disjoint and $f(S)=f_1(S\cap X_1)+f_2(S\cap X_2)$. Here $f_1$ and $f_2$ are submodular on $X_1$ and $X_2$ respectively.
Here $X_1,X_2,f_1,f_2$ are unknown and only a value query access to $f$ is given. Then is there a polytime algorithm which finds $X_1$. If there are multiple choices for $X_1$ any of them should be fine.
Some thoughts. If we can find any two elements $t_1,t_2$ such that both either belong to $X_1$ or belong to $X_2$ then we can merge them and proceed recursively. But it is not clear how to implement such a step.