Existential closedness conjecture for the modular $j$-function and its derivatives

Let $J:\operatorname{\mathbb{H}}\to\operatorname{\mathbb{C}}^3$ be defined by

$$J(z)=(j(z),j'(z),j”(z)),$$

and extend it coordinatewise to $\operatorname{\mathbb{H}}^n$, with graph $\Gamma_J\subseteq\operatorname{\mathbb{H}}^n\times\operatorname{\mathbb{C}}^{3n}$. A variety is called $J$-broad, $J$-free, or $\operatorname{\mathbb{H}}$-free according to the properties defined in the source. Existential closedness conjecture for $J$. If $V\subseteq\operatorname{\mathbb{H}}^n\times\operatorname{\mathbb{C}}^{3n}$ is an irreducible $J$-broad, $J$-free and $\operatorname{\mathbb{H}}$-free variety defined over $\operatorname{\mathbb{C}}$, then

$$V\cap\Gamma_J\neq\emptyset.$$

This extends the existential-closedness expectation from the modular $j$-function to its first two derivatives; higher derivatives are algebraic over these and therefore do not add transcendence-theoretic information. The source proves related results for blurred $j$-functions and derivatives, but does not resolve this conjecture.