On the DLW conjectures

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture 1 is a claim about the uniqueness of certain monomial graphs. Conjecture 2, which implies Conjecture 1, deals with certain permutation polynomials of finite fields. Two natural strengthenings of Conjecture 2, referred to as Conjecture A, Conjecture B in the present paper, were also insinuated. In a recent development, Conjecture 2 and hence Conjecture 1 have been confirmed. The present paper gives a proof of Conjecture A.