Do all liaison-constructed (5, 2)-geprofi sets lie on an elliptic quintic curve in P4?

Ascertain whether every set Z ⊂ P4 of 10 points in linear general position with h-vector (1,4,4,1), obtained by linking a general set of six points in a hyperplane via four quadric hypersurfaces as in Example 5.11, lies on a nondegenerate smooth degree-5 genus-1 curve (an elliptic quintic) in P4.

Background

The paper presents two families of (5, 2)-geprofi sets in LGP: those on rational normal quartics and those constructed by liaison, where four quadrics link a general set of six points in a hyperplane to a residual set Z of ten points having h-vector (1,4,4,1).

While some such Z arise as degree-2 divisors on smooth elliptic quintic curves (degree 5, genus 1), the authors explicitly state that it is unknown whether every Z built by the liaison construction lies on such a curve.