The prime-values conjecture for Drinfeld-module auxiliary polynomials

Let $A$ be the coefficient ring and let

$$\phi:A\rightarrow A\{\tau\}$$

be a Drinfeld module. For each prime polynomial $P$, let $g_{P,\phi}$ denote the auxiliary polynomial associated with $P$ and $\phi$. Prime-values conjecture. There exist infinitely many prime polynomials $P$ such that $g_{P,\phi}$ is prime. This conjecture is stated as the remaining assumption needed to prove that infinitely many values $\phi_P(a)$ are non-prime, or equivalently that infinitely many $\phi$-Mersenne numbers are not prime; the supplied text gives no resolution.