The stable forking conjecture over a base

Let $T$ be a simple $\mathcal{L}$-theory, with $a$, $b$, and $C$ tuples or sets in a sufficiently saturated model. 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 is a stable $\mathcal{L}(C)$-formula when it is stable in the language expanded by constants for $C$.

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

This is presented as a weakening of the original stable forking conjecture, and the source says that it remains very much open. It agrees with the original formulation over the empty base after expanding the language by constants for the base.