(This is a followup to my previous question on rationally connected varieties)

Let $X$ be a smooth hypersurface of degree $d$ in $\mathbb P^n$, $n \geq 3$.

Let $f: Y \to X$ be a finite morphism from a normal variety $Y$ to $X$, generically of degree $k$. Let $D$ be the branch divisor of $f$.

Do we have for some constant $c_{d,n}$ the inequality $$ H^{n-2} \cdot D \geq (c_{d,n}-o(1)) k?$$

As $n \geq 3$, by Lefschetz $X$ is simply connected, so clearly $D$ is nonzero and hence $H^{n-1} \cdot D \geq 1$. I'm asking if the inequality can be strengthened.

Jason Starr's positive answer to my previous question, plus the fact that Fano varieties are rationally connected, shows the answer is yes when $d < n+1$.

But I don't know anything about the Calabi-Yau case $d=n+1$ or the general type case, except that by pulling back covers from projective space you can see that $c_{d,n} \leq 2 d$ if it exists.

If the answer is yes, the only strategy I can see to prove it would be some Lefschetz type argument where you show that a cover extends. But it's certainly not the case that a ramified cover of a hypersurface in projective space always is a pullback of a ramified cover of projective space - then the branch divisor would always be a multiple of the hyperplane class.