The omega-categoricity conjecture for second-order strongly minimal sets

Let $\mathcal{X}$ be a strongly minimal set of differential order $2$ definable over the field of constants $\mathcal{C}$. A set is $\omega$-categorical if, for every non-constant solution $x\in\mathcal{X}$, the set $\mathcal{C}\langle x\rangle^{\mathrm{alg}}\cap\mathcal{X}$ is finite. For distinct non-constant solutions $x_1,\ldots,x_n$ satisfying

$$\operatorname{tr.deg}_{\mathcal{C}}\mathcal{C}\langle x_1,\ldots,x_n\rangle=2n,$$

The $\omega$-categoricity conjecture. The set $\mathcal{X}$ is $\omega$-categorical if and only if there is a positive integer $k$ such that, for every such tuple $x_1,\ldots,x_n$, all but $kn$ other non-constant solutions $x$ satisfy

$$\operatorname{tr.deg}_{\mathcal{C}}\mathcal{C}\langle x_1,\ldots,x_n,x\rangle=2n+2.$$

This conjecture concerns the proposed characterization of finite algebraic dependence among solutions of second-order strongly minimal differential equations. In the stated source, the general conjecture is disproved by a counterexample of Freitag and Scanlon arising from the third-order differential equation satisfied by the $j$-function.