Does de Franchis' theorem hold over any base field

Question

Let $k$ be a field and let $X$ be a hyperbolic curve over $k$.

Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$.

I know this statement holds over $k=\mathbf{C}$. In particular, it holds over $k=\overline{\mathbf{Q}}$.

Does it hold over any field $k$?

What about a number field?

Answer 1

The answer is yes if your curves are projective, see E. Kani: Bounds on the number of nonrational subfields of a function field. Invent. Math. 85 (1986), no. 1, 185–198.

Over an algebraically closed field, a hyperbolic curve is either a non-empty open subset of a projective curve of genus $g\ge 2$, or an elliptic curve minus at least one point, or the projective line minus at least three points. Equivalently, this is a smooth geometrically connected curve (not necessarily projective) with finite automomorphism group.