Finite-parameter determination of invariant types

Let $p(x)$ be a global $M$-invariant type and let $\Delta$ be a finite set of formulas. Finite determination conjecture. There is a finite set $\Delta'$ of formulas such that, for every $\phi(x;y)\in\Delta$ and all $b,b'\in\mathcal U$, if

$$\operatorname{tp}_{\Delta'}(b/M)=\operatorname{tp}_{\Delta'}(b'/M),$$

then

$$p\vdash \phi(x;b)\leftrightarrow\phi(x;b').$$

A positive answer to the preceding invariant-type quotient conjecture would imply this statement; it is therefore presented as an open consequence/question.