Determine the very general degree of irrationality in F1,3

Determine whether a very general elliptic surface S in the moduli space F1,3 of relatively minimal elliptic surfaces with a section, base genus one, and holomorphic Euler characteristic three has degree of irrationality irr(S)=4.

Background

The paper proves that a very general member of Fa,b has degree of irrationality at least four when the base curve has genus at least two. It also constructs special elliptic surfaces with smaller degree of irrationality, including examples with chi(OS)=3 and degree three.

The case F1,3 is singled out because |L| is isomorphic to P2; a degree-three rational map would therefore induce a self-correspondence between two projective planes. The paper does not determine whether the very general member instead has degree exactly four.