Low-degree factorization conjecture for polynomial congruences

Let $q$ be a power of a prime $p>2$, let $d$ be a natural number, and let $h\in\mathbb{F}_q[x]$ be relatively prime to $x$ with $\deg h=r<d/4$. Suppose there are $g_1,g_2\in\mathbb{F}_q[x]$ such that

$$h\equiv g_1(x^p)g_2(x^2)\pmod{x^d}.$$

Low-degree factorization conjecture. For all sufficiently large $d$, there exist $g_3,g_4\in\mathbb{F}_q[x]$ such that

$$h=g_3(x^p)g_4(x^2).$$

In particular, if $h$ is irreducible, then $h=g_4(x^2)$. Proving this conjecture would settle the corresponding case $r<d/4$ in the odd-family trace problem; it is motivated by heuristics concerning irreducible polynomials in the relevant residue subgroups.