Maximum rainbow solutions to x plus y less than z

Determine asymptotically the maximum number of rainbow solutions to x + y < z with x ≤ y over all 3-colorings of [1,n], and prove or disprove the conjecture that the asymptotic value is the constant given in Theorem 5.7, namely (3√3−6)n^3+O(n^2).

Background

Theorem 5.7 determines the maximum number of rainbow solutions within the restricted family of 3-colorings whose color classes are single intervals, obtaining (3√3−6)n^3+O(n^2). The authors state that they have made little progress on the unrestricted problem and provide only the bounds n^3/31(1+o(1)) < RM^3(n) ≤ n^3/27(1+o(1)). They conjecture that the restricted-family value from Theorem 5.7 is nevertheless the correct unrestricted asymptotic value.