Generalizations of de Franchis and function field Mordell The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," Inventiones, 1975), states that if $X$ is a complex projective varity of general type, then for any fixed complex projective variety $Y$, there are only finitely many surjective mappings $Y \to X$. (A variant of this for Kobayashi-hyperbolic complex projective manifolds $X$ was conjectured by Lang and proved by C. Horst; those manifolds are expected to be of general type anyway, but if I understand correctly, this expectation has not been proved in full generality.)
I would like to ask here about the generalization of this problem whereby we drop the surjectivity assumption, and assume instead that $Y \to X$ is generically finite onto its image. To keep things simple, consider $\dim{Y} = 1$ and $\dim{X} = 2$, which is the first interesting case:
Question. Let $X$ be a surface of general type and $Y$ a fixed curve. Assume that there is no curve $Z$ such that $X$ is rationally dominated by $Y \times Z$. Does $X$ contain only finitely many images of non-constant morphisms $Y \to X$? [Is it plausible that the manifestly necessary assumption on $X$ should be sufficient?] 
This would be a generalization of the function field Mordell conjecture. [To see this, let $X \to B$ be a fibred surface of genus $> 1$, and choose any branched covering $Y \to B$ of genus $> 1$. Then $X \times_B Y$ is a surface of general type, and if it is domianted by a product $Y \times Z$, then $X/B$ is isotrivial. ] It would likewise generalize the theorem of Miyaoka and Lu (itself a very particular case of Lang's geometric conjecture), according to which a surface of general type contains only finitely many smooth rational curves.
I am aware of a 1985 paper by Y. Imayoshi ("Holomorphic maps of compact Riemann surfaces into $2$-dimensional compact $C$-hyperbolic manifolds," Math. Ann. 270), which solves the analogous question for a compact complex surface $X$ which admits a covering $\Omega \to X$ such that the bounded holomorphic functions on $\Omega$ separate points (i.e., the Caratheodory pseudo-distance is a distance on $\Omega$; such $X$ are a fortiori Kobayashi-hyperbolic.) In that case the conclusion is in fact the finiteness of the set of non-constant holomorhic maps from a fixed compact hyperbolic Riemann surface, and it is enough to assume that $X$ does not have a finite etale cover by the product of two compact hyperbolic Riemann surfaces.
But presumably, $C$-hyperbolicity is a very restrictive constraint on a surface of general type. Have the above algebraic statement or similar strengthenings of function field Mordell been considered, or at least explicitly stated anywhere? 
 A: One of the major results in this direction, when $Y$ is either a rational or an elliptic curve, is due to Bogomolov, see 
F. Bogomolov, Families of curves on a surface of general type, Doklady AN SSSR 236 (1977),
no. 5, 1041-1044, in Russian; English translation: Soviet Math. Dokl. 18 (1977), 1294-1277.
The statement of Bogomolov's result is as follows:

Theorem (Bogomolov). Let $X$ be a surface of general type with $c_1^2(X) > c_2(X)$. Then, for any $g$, the curves of geometric genus $g$ on $X$ form a bounded family (roughly speaking, a family having only a finite number of irreducible components).

Since $X$ is of general type, it is not covered by rational or elliptic curves, i.e. curves of geometric genus $0$ and $1$ cannot deform. Then Bogomolov's theorem implies that there are only a finite number of rational or elliptic curves on $X$, that is we obtain the following

Corollary. Let $X$ be a surface of general type with $c_1^2(X) > c_2(X)$, and let $Y$ be either a rational curve or an elliptic curve. Then there are only finitely many images of non-constant morphisms $ Y \to X$. 

The condition $c_1^2(X) > c_2(X)$ is rather restrictive, for instance it is never satisfied for a hypersurface $X \subset \mathbb{P}^3$. It is expected that in any case neither rational nor elliptic curves can be dense on a variety of general type $X$ (I think this is a conjecture due to Kobayashi), but in principle one could have surfaces  of general type with $c_1^2(X) \leq c_2(X)$ and containing countably many rational or elliptic curves. However, I'm not aware of any example of this type.
