On a conjecture on planar polynomials of the form X(Trn(X)−uX)

In a recent paper, Kyureghyan and Özbudak proved that u∈{1,2} was a sufficient condition for the polynomial X(Xq2+Xq+(1−u)X) to be planar over Fq3, and conjectured the condition was also necessary. This conjecture is established in this note.