The Zilber–Pink conjecture for curves in powers of the modular curve

Let $Y(1)$ denote the modular curve, and let $S\subset Y(1)^n$ be an irreducible curve that is not contained in a proper special subvariety of $Y(1)^n$. For a special subvariety $Z\subset Y(1)^n$, write $\operatorname{codim} Z$ for its codimension. Zilber–Pink conjecture. The set

$$S(\mathbb{C})\cap \bigcup_{\operatorname{codim} Z\geq 2}Z(\mathbb{C})$$

is finite, where the union ranges over all special subvarieties $Z$ of $Y(1)^n$ of codimension at least $2$. This is the predicted finiteness of atypical intersections for curves in powers of the modular curve; the stated special case is open without additional assumptions on $S$.