Prime power conjecture for cyclic projective planes

Prove or refute that no cyclic projective planes exist for any order that is not a prime power; equivalently, demonstrate that perfect difference sets modulo v = q^2 + q + 1 exist only when q is a prime power.

Background

Singer’s construction shows that cyclic projective planes (equivalently, perfect difference sets modulo v = q^2 + q + 1) exist when q is a prime power. The prime power conjecture posits nonexistence for all other orders.

Computational evidence has verified nonexistence up to large bounds (at least two billion), but a general proof is still lacking.