Minimum rainbow Schur solutions in linearly exact 3-colorings

Determine asymptotically the minimum number of rainbow solutions to x + y = z over all δn-exact 3-colorings of [1,n].

Background

The paper studies Gallai-Schur multiplicities and introduces δn-exact colorings, in which each of the three colors is used at least δn times. It does not determine the asymptotic minimum number of rainbow solutions to the Schur equation x+y=z under this linear exactness condition, and lists this as a separate open problem.