Definable $(p,q)$-conjecture for formulas of VC-codensity less than $q$

Let $q\geq 2$ be an integer, $M$ be an $L$-structure, and let $\varphi(x,y)$ be an $L(M)$-formula with dual shatter function $\pi_{\varphi}^*(n)\in o(n^q)$. A family of instances $\{\varphi(x,a):a\in A\}$ has the $(p,q)$-property if, among any $p$ instances, some $q$ have nonempty intersection. Definable $(p,q)$-conjecture. If there exists an integer $p\geq q$ such that $\varphi(x,y)$ has the $(p,q)$-property, then there exists some $m<\omega$ and $L(M)$-formulas $\psi_1(y),\ldots,\psi_m(y)$ such that, for every $i\leq m$, the family

$$\{\varphi(x,a):a\in\psi_i(M)\}$$

is consistent. This is the definable model-theoretic analogue of the Alon–Kleitman–Matoušek $(p,q)$-theorem and strengthens the base case of the definable $(p,q)$-conjecture; the source notes that the version commonly found in the literature assumes the stronger hypothesis that the whole structure is NIP.