Liftings with n=2 beyond even-dimensional Singer complements

Prove or disprove the conjecture that only difference sets with parameters of complements of Singer difference sets with even dimension admit liftings to relative difference sets with n=2.

Background

The paper cites work of Arasu, Jungnickel, Ma, and Pott showing that several families of difference sets—including Paley, twin-prime-power, and classical Singer-parameter difference sets—cannot have liftings with n=2. Against this background, the cited authors proposed a classification conjecture for all difference sets admitting such liftings.

The conjecture restricts possible n=2 liftings to complements of Singer difference sets in even dimension, making it a nonexistence-and-classification problem for cyclic relative difference sets.