The two-factor conjecture for iterated $Q_c$-transforms

Let $f$ be an irreducible polynomial over $\mathbb F_q$ of degree $n\ge 3$, and let $Q_c$ be a polynomial of degree $D$. Suppose that $n$ is odd and $D\equiv 2\pmod 4$, and that the $Q_c$-transform $f^{Q_c}$ is irreducible. Two-factor conjecture. Then $(f^{Q_c})^{Q_c}$ factors over $\mathbb F_q$ into two irreducible polynomials, each of degree $\frac{nD^2}{2}$. The claim is motivated by computational tests and concerns the exceptional case in which the general irreducibility result does not apply; it remains open in the supplied source.