Ax--Schanuel conjecture for variations of mixed Hodge structure

Let $X$ be a smooth connected algebraic variety over $\mathbb{C}$, with base point $b \in X(\mathbb{C})$. Let $\mathscr{H}_{\mathbb{Z}}$ be a $\mathbb{Z}$-variation of mixed Hodge structure on $X$, with generic Mumford--Tate group $\operatorname{\mathbf{MT}}_{\mathscr{H}_{\mathbb{Z}}}$. After passing to a finite cover if necessary, let $\Lambda$ be the image of the monodromy representation in $\operatorname{\mathbf{MT}}_{\mathscr{H}_{\mathbb{Z}}}(\mathbb{Z})$, let $G$ be its $\mathbb{Q}$-Zariski closure, and let $D=D(G)$ be the associated weak mixed Mumford--Tate domain. Let $\varphi\colon X(\mathbb{C})\to\Lambda\backslash D$ be the period map, let $\check{D}$ be the compact dual of $D$, let $V\subseteq X\times\check{D}$ be algebraic, and let $W$ be an irreducible analytic component of $V\cap\Lambda$ whose projection to $X$ is not contained in any proper weakly special subvariety. Ax--Schanuel conjecture for variations of mixed Hodge structure. One has

$$\operatorname{codim}_{X\times\check{D}}(W)\geq\operatorname{codim}_{X\times\check{D}}(V)+\operatorname{codim}_{X\times\check{D}}(\Lambda).$$

The conjecture is the key conditional input for the paper's functional-transcendence and higher-dimensional Chabauty--Kim results. The source notes that for $n=1$ it specializes to the Ax--Schanuel theorem for abelian varieties, while the general statement is used conditionally and is not resolved here.