The BIG condition for Mumford–Tate varieties with infinitely many special curves

Let $X$ be an irreducible Mumford–Tate variety associated to a group $G$, and suppose that $X$ contains infinitely many special curves. For finitely many compactified special curves $C_i$ in $X$, let $C_i^0$ denote their open parts and consider the monodromy representation of

$$\pi_1\left(\bigcup_{i \in I} C_i^0,y\right).$$

The BIG condition conjecture. The ${\mathbb Q}$-Zariski closure of this monodromy representation is equal to $G$. Equivalently, condition (BIG) holds for $X$. This conjecture would provide the monodromy input needed in the paper's strategy for proving specialness criteria for horizontal algebraic subvarieties of Mumford–Tate varieties; its resolution is not given in the source.