Modular Ax-Schanuel conjecture with derivatives

Let $A$ be as in the complex Modular Ax-Schanuel setting, with geodesically independent coordinates $z_1,\ldots,z_n$ and the modular function $j$. Modular Ax-Schanuel conjecture with derivatives. If the $z_i$ are geodesically independent, then

$$\operatorname{trdeg}_{\mathbb{C}}\mathbb{C}(z_i,j(z_i),j'(z_i),j”(z_i))\geq 3n+\dim A.$$

The conjecture strengthens the complex Modular Ax-Schanuel statement by including derivatives of $j$. The source notes Mahler's algebraic independence theorem for $j,j',j”$ and the differential-algebraic dependence of $j”'$, but gives no resolution of the full conjecture.