Invariant-type quotient conjecture

Assume that $T$ is NIP. Let $M\prec^+N$, let $a\in\mathcal U$ be such that $\operatorname{tp}(a/N)$ is $M$-invariant, and let $\phi(x;y)\in L(M)$.

Invariant-type quotient conjecture. There are $b\in\mathcal U$ and a formula $\psi(z;y)$ such that $\operatorname{tp}(b/N)$ is finitely satisfiable in $M$ and

$$\phi(a;N)=\psi(b;N).$$

This is posed as a test question about analysing an invariant type through a finitely satisfiable type and a definable quotient. The supplied text gives no resolution.