Existence of characteristic-zero geproci sets in dimension greater than three

Determine whether there exists a finite geproci set in $P^n$ for some $n>3$ over an algebraically closed field of characteristic zero.

Background

The paper constructs infinitely many geproci sets in every projective dimension $n\geq3$ over algebraically closed fields of positive characteristic. The construction fundamentally uses finite additive subgroups $F_{p^e}$ and Artin–Schreier polynomials.

The authors note that this mechanism has no direct characteristic-zero analogue, but do not resolve whether finite geproci sets in dimensions greater than three exist in characteristic zero by other methods.