Amerik's boundedness conjecture for finite morphisms

Let $f:Y\to X$ be a finite morphism between smooth projective varieties of dimension $n\ge 2$, each having second Betti number $1$. Suppose that $X\not\cong\mathbb{P}^n$. Amerik's boundedness conjecture. The degree $\deg(f)$ is bounded in terms of the discrete invariants of $Y$ and $X$. The conjecture is known when $\dim(X)\le 3$ or when $X$ is a smooth quadric of dimension at least $3$; the general case remains open.