Definability in families of Kolchin invariants

Let $\langle \mathfrak C,\bar \delta\rangle$ be a monster model of $T^{\bar \delta}_g$. For an $L^{\bar \delta}$-definable set $X$, let $\omega_X$ be the supremum of the relevant Kolchin polynomials and let $\mu(X)$ be the supremum of their leading monomials. Kolchin-invariant definability conjecture. For every $L^{\bar \delta}$-definable family $(X_i:i\in I)$, there is a partition $I=I_1\sqcup\dotsb\sqcup I_m$ into finitely many definable sets such that $\mu(X_i)$ and $\omega_{X_i}$ are constant on each $I_j$. This would make the Kolchin polynomial and its leading monomial behave uniformly in definable families; the source cites related work and gives no resolution.