The stable forking conjecture

Let $T$ be a simple $\mathcal{L}$-theory, and let $a$, $b$, and $C$ be tuples or sets in a sufficiently saturated model of $T$. Write $a \mathrel{\mathop{\mathpalette\notind{}}_{C}} b$ for forking dependence over $C$, and let $\operatorname{tp}(a/Cb)$ denote the type of $a$ over $Cb$. A formula $\varphi(x,y)$ is stable when it does not have the order property.

Stable forking conjecture. If $a \mathrel{\mathop{\mathpalette\notind{}}_{C}} b$, then there is a formula $\varphi(x,\overline{b})\in\operatorname{tp}(a/Cb)$ such that $\varphi(x,\overline{b})$ forks over $C$, and such that $\varphi(x,y)$ is stable.

This conjecture proposes that every forking dependence instance in a simple theory is witnessed by a stable formula. Its status is not resolved in the source.