Non-zero coefficients of primitive polynomials

Question

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$ be positive integers $\geq 2$. I want to prove that there exists a primitive polynomial $$F(x) = x^{mn}-\sum\limits_{j=0}^{mn-1}f_jx^j\in R[x]$$ with the property that there is a $k\in\{1,2,\ldots,n-1\}$ such that $f_{km}\neq 0$. I think that this is always true except in case $q=m=n=2$.

Answer 1

I post here a Hansen-Mullen's conjecture, which was proved in 2007. The validity of my assumption in the question is a straightforward consequence of this conjecture.

Conjecture.(Hansen and Mullen, 1992) Let $a\in \mathbb{F}_q$ and let $n\geq2$ be a positive integer. Fix and integer $m$ with $0<m<n$. Then there exists a primitive polynomial $f(x) = x^n +\sum_{j=1}^{n}a_jx^{n-j}$ over $\mathbb{F}_q$ with $a_m = a$ with (genuine) exceptions when $$ (q,n,m,a) = (q,2,1,0), (4,3,1,0), (4,3,2,0), (2,4,2,1). $$ The proof is carried out in several stages. See for details Stephen D Cohen, Primitive polynomials with a prescribed coefficient, Finite Fields and Their Applications 12 (2006) 425-491. The completion of the proof is given in Cohen and Presern, The Hansen-Mullen primitivity conjecture: completion of proof, in McKee and Smyth, eds., Number Theory and Polynomials, London Mathematical Society Lecture Note Series 352.