Classification conjecture for power difference sets over prime fields

Prove or disprove that if the multiplicative subgroup of \(\ell\)-th powers in \(F_p^\times\) forms a difference set of the prime field \(F_p\), then \(\ell\) must equal \(2\), \(4\), or \(8\).

Background

For prime fields, known results establish the existence of power difference sets for indices $2$, $4$, and $8$ under suitable arithmetic conditions, while nonexistence has been proved for several larger indices, including $6$, $10$, $12$, $14$, $16$, $18$, $20$, $22$, $24$, $26$, and $32$.

The conjecture asserts that these known indices exhaust all possibilities. The paper seeks structural conditions using cyclotomic matrices rather than deriving separate explicit formulas for every index, but leaves the conjecture unresolved.