Establish an analogue of the Graber–Harris–Starr theorem for connecting-gonality coverings

Determine whether an analogue of the Graber–Harris–Starr theorem holds for a dominant diagram in which a variety Y dominates a smooth projective variety X and maps generically finitely to a rationally connected variety R, as arising in the characterization of the eventual p-connecting gonality of X.

Background

The paper characterizes the eventual p-connecting gonality of a smooth projective variety X through diagrams in which a variety Y dominates X and admits a generically finite morphism to a rationally connected variety R. The degree of this morphism controls the gonality of curves connecting arbitrarily many general points of X.

The authors leave unresolved whether the Graber–Harris–Starr theorem, which concerns the existence of sections in families over curves, has an analogue in this covering and connecting-gonality setting.