Strong minimality conjecture for generic differential equations

Let $f(x)$ be a generic differential polynomial of fixed order $h>1$ and fixed degree $d>1$, meaning that its coefficients are independent differentially transcendental elements and it contains all monomials of degree at most $d$ in $x,x',\ldots,x^{(h)}$. Let $Z(f)$ denote its solution set. Strong minimality conjecture. Generic differential equations of fixed order and degree greater than one are strongly minimal. The main theorem proves strong minimality under a sufficiently large degree bound, while this conjecture predicts the result for every degree greater than one.