The Ax-Schanuel conjecture for enlarged mixed Shimura varieties

Let $S^\natural$ be a connected enlarged mixed Shimura variety associated with $(P,\mathcal{X}^{\natural+})$, with uniformization $\mathrm{unif}^\natural\colon\mathcal{X}^{\natural+}\to S^\natural$. Let $\widetilde{Z}^\natural$ be a complex analytic irreducible subvariety of $\mathcal{X}^{\natural+}$, let $Z^\natural=\mathrm{unif}^\natural(\widetilde{Z}^\natural)$, and let $\mathbscr{Z}^\natural$ be the graph of this restriction. Let $F^\natural$ and $\widetilde{F}^\natural$ be the smallest bi-algebraic, equivalently quasi-linear, subvarieties containing $Z^\natural$ and $\widetilde{Z}^\natural$, respectively, so that $F^\natural=\mathrm{unif}^\natural(\widetilde{F}^\natural)$. Write $\widetilde{X}^\natural=(\widetilde{Z}^\natural)^{\mathrm{Zar}}$, $Y^\natural=(Z^\natural)^{\mathrm{Zar}}$, and $\mathbscr{B}^\natural=(\mathbscr{Z}^\natural)^{\mathrm{Zar}}$. Ax-Schanuel conjecture for enlarged mixed Shimura varieties. One has

$$\dim\widetilde{X}^\natural+\dim Y^\natural-\dim\widetilde{Z}^\natural\geqslant\dim F^\natural.$$

Moreover, with $\mathbf{pr}^{\mathrm{lin}}_{F^\natural}$ defined from the linear projections of $\widetilde{F}^\natural$ and $F^\natural$ as in the source,

$$\dim\mathbf{pr}^{\mathrm{lin}}_{F^\natural}(\mathbscr{B}^\natural)-\dim\mathbf{pr}^{\mathrm{lin}}_{F^\natural}(\mathbscr{Z}^\natural)\geqslant\dim(F^\natural)^{\mathrm{lin}}.$$

The conjecture is proposed because the naive Ax–Schanuel dimension inequality fails in the presence of the nonlinear part of $F^\natural$; the supplied text gives no resolution.