The stronger polynomial selection conjecture for small-characteristic DLP

Let $\mathbb F_q$ be a finite field of odd characteristic. The polynomials $h_1,h_2\in\mathbb F_{q^d}[X]$ are required to be coprime and have degree at most $2$. Stronger polynomial selection conjecture. There exists an integer $d=O(\log(q))$ and such polynomials $h_1,h_2$ such that, for every positive integer $\ell\leq\deg(h_1)+q$, there is an element $t_0\in\mathbb F_{q^d}$ for which $h_1X^q+h_2-t_0$ has an irreducible factor of degree $\ell$. This stronger requirement is motivated by the use of polynomials of the form $h_1=1$ and $h_2=X^2-t_0$ in the discrete logarithm algorithm; its proof would support a non-heuristic quasi-polynomial-time algorithm in small characteristic.