Existential closedness for reducts of differentially closed fields with the j-function

Let $\mathfrak{L}_{j}$ denote the language of the reduct associated with the differential equation of the modular $j$-function, and let $\emph{EC}$ denote the existential-closedness axiom scheme referred to in the source. Existential-closedness conjecture. ($\mathfrak{L}_{j}$-reducts of) differentially closed fields satisfy $\emph{EC}$.

The source explains that this would follow from adequacy of the Ax–Schanuel inequality for $j$ and would provide a complete axiomatisation with near model completeness. The author describes adequacy as difficult and leaves this assertion as a conjecture.