Existence conjecture for irreducible polynomials with prescribed finite-field coefficients

Let $q$ be a prime power, and let $k,l$ be positive integers. Write $c\mathcal{F}_k(q)$ for the set of elements of degree $k$ over $c\mathbb{F}_q$. Existence conjecture. There exists a monic irreducible polynomial $f(x)cinc\mathbb{F}_{q^k}[x]$ of degree $l$ such that $f(x)-f(0)cinc\mathbb{F}_q[x]$ and $f(0)cinc\mathcal{F}_k(q)$. This strengthens the preceding proposition because $l$ need not be the smallest prime divisor of $kl$; the conjecture is presented without a resolution in the source.