Ax–Schanuel conjecture for variations of mixed integral Hodge structures

Let $(S,\mathbb{V})$ be a $\mathbb{Z}$VMHS, let $U\subset\widetilde S\times S$ be an algebraic subvariety, and let $W$ be an irreducible component of $U\cap\Delta$, where $\Delta$ is the graph of $\pi:\widetilde S\longrightarrow S$. Ax–Schanuel for $\mathbb{Z}$VMHS. Then

$$\operatorname{cd}_U W\geq \dim \overline{W}^{\mathrm{ws}},$$

where $\overline{W}^{\mathrm{ws}}$ denotes the smallest weakly special subvariety of $S$ containing $\pi(W)$. This is identified as the main functional-transcendence conjecture for variations of mixed integral Hodge structures; no resolution is given in the source.