Extensible definability conjecture for non-forking formulas

Let $T$ be NIP and $M\models T$. Let $\phi(x;d)\in L(\mathcal U)$ be a formula non-forking over $M$.

Extensible definability conjecture. There is $\theta(y)\in \operatorname{tp}(d/M)$ such that the partial type

$$\{\phi(x;d'): d'\in \theta(\mathcal U)\}$$

is consistent.

This is presented as the central conjecture of the paper and first appeared in earlier work. Its status is not resolved in the supplied text.