The adequacy conjecture for the $j$-function's Ax–Schanuel inequality

Let $C$ be an algebraically closed field with $\operatorname{td}(C/\mathbb{Q})=\aleph_0$, let $\mathfrak{C}$ be the class of $E_j$-fields with constant field $C$, and let $U$ be its strong Fraïssé limit. A differentially closed field is a field satisfying the theory of differential closure; its $\mathfrak{L}_j$-reduct, denoted $\mathcal{K}_{E_j}$, is the structure obtained by retaining the $j$-function relation. Adequacy conjecture. If $\mathcal{K}$ is a countable saturated differentially closed field, then

$$\mathcal{K}_{E_j}\cong U.$$

In the terminology attributed by the source to Aslanyan, this says that the Ax–Schanuel inequality for the differential equation of the $j$-function is strongly adequate. The conjecture is equivalent to the claim that these reducts satisfy existential closedness.