Let $X$ be a smooth projective variety with negative Kodaira dimension over $\mathbb{C}$.

Is there an integer $n\geq 1$, a smooth projective hyperkahler variety $H$, and a finite morphism $H\to X^n$?

Is the answer positive for Fano varieties?

Motivation: Varieties $X$ with negative Kodaira dimension should admit an entire curve $\mathbb C\to X(\mathbb{C})$. If the answer to the above question is positive, we reduce this to the case of hyperkahler.

Guess: The answer is probably negative. But what's the easiest example?