Zilber-Pink in $Y(1{)}^n$: Beyond multiplicative degeneration

We establish Large Galois orbits conjectures for points of unlikely intersections of curves in $Y(1{)}^n$, upon assumptions on the intersection of such curves with the boundary $X(1{)}^n\mathrm{\backslash }Y(1{)}^n$, in the Zilber-Pink setting. As a corollary, building on work of Habegger-Pila and Daw-Orr, we obtain new cases of the Zilber-Pink conjecture for curves in $Y(1{)}^n$.