The algebraic-independence characterization of general type equations

Let $k$ be a differential field with field of constants $C$. A first order differential equation over $k$ and an autonomous differential equation over $C$ are said to be of general type according to the classification used in the paper. Consider differential field extensions of $k$, respectively $C$, whose field of constants is $C$, and algebraically independent solutions in those extensions.

General type characterization conjecture. A first order differential equation over $k$ (respectively, an autonomous differential equation over $C$) is not of general type if and only if it has at most three (respectively, at most one) algebraically independent solutions in any given differential field extension of $k$ (respectively, $C$) having $C$ as its field of constants.

This conjecture is motivated by preceding results showing that a first order equation with a nonalgebraic solution in an iterated strongly normal extension forces any four nonalgebraic solutions to be $k$-algebraically dependent, while the analogous autonomous result gives algebraic dependence for any two solutions over $C$.