Kodess's isomorphism conjecture for monomial digraphs

Let $D(q;m,n)=D_2(\mathbb F_q;X^mY^n)$ be a monomial digraph over the field with $q$ elements. Kodess's conjecture. For a prime power $q$, the digraphs $D(q;m_1,n_1)$ and $D(q;m_2,n_2)$ are isomorphic if and only if there exists an integer $k$ coprime to $q-1$ such that

$$m_2\equiv km_1\pmod{q-1},$$

and

$$n_2\equiv kn_1\pmod{q-1}.$$

The sufficiency is described as easy to verify, while necessity remains to be established.