Independence characterization for sufficient reducts

Let $T_W$ be a theory in a signature $L_W$, with reducts $T_w$ to a sufficient collection of subsignatures $L_w$. Assume that $M_W$ is a monster model of $T_W$, that $T_W$ and every $T_w$ are simple (respectively rosy), and that $\mathop{\mathpalette\Ind{}}^W$ and $\mathop{\mathpalette\Ind{}}^w$ denote forking (respectively thorn-forking) independence. Let $C\subset A,B\subset M_W^{eq}$ be algebraically closed in the sense of $T_W^{eq}$.

Independence characterization conjecture. Then

$$A \mathop{\mathpalette\Ind{}}^W_C B \text{ if and only if } A \mathop{\mathpalette\Ind{}}^w_C B \text{ for all } w.$$

The claim seeks to characterize independence in the full theory using independence in all reducts. The source gives related one-way results and notes that the converse is immediate in stable theories, while the general simple or rosy case remains conjectural.