Distinguish the displayed nontrivial Gauss-sum equalities after twisting

Determine whether, for the examples with q=p^{2n} arising from factorizations p^n+1=c_1M_1=c_2M_2 with c_1 and c_2 even, the equal Gauss sums G(\chi_1)=G(\chi_2)=p^n associated with characters of orders M_1 and M_2 become unequal after twisting by the lift of a character of order p-1.

Background

The paper constructs nontrivial Gauss-sum equalities using Stickelberger’s evaluation. Specifically, when p^n+1=c_1M_1=c_2M_2 and q=p^{2n}, characters \chi_j of orders M_j can satisfy G(\chi_1)=G(\chi_2)=p^n. If M_1\neq M_2, these characters cannot be Frobenius-conjugate, so the equality is genuinely nontrivial.

The authors note that twisting by the lift of a character of order p-1 may separate these equal Gauss sums, but they do not establish this for the displayed examples. The unresolved issue is therefore whether the twisted Gauss sums are unequal in those concrete cases.