Isomorphism classification of monomial digraphs

Prove or disprove that, for every prime power q, the monomial digraphs D(q;m1,n1) and D(q;m2,n2) are isomorphic if and only if there exists an integer k coprime to q−1 such that m2≡km1 (mod q−1) and n2≡kn1 (mod q−1).

Background

The paper states that the sufficiency of the arithmetic condition is easy to verify, but its necessity remains unproved. It emphasizes that the conjecture is unresolved even when q is prime.