Existence conjecture for ordinary rank-2 Drinfeld modules with prescribed Frobenius matrix

Let $A$ be the coefficient ring, let $\Phi$ be a rank-$2$ Drinfeld $A$-module with characteristic ideal $\chi_{\Phi}$, and let $P$ be the characteristic polynomial datum appearing in the Frobenius construction. Write

$$\overline{P}=P\pmod{\chi_{\Phi}}.$$

Let $M\in\mathbf{M}_{2}(A/\chi_{\Phi})$ satisfy

$$\det(M)=\overline{P}^{m},\qquad \operatorname{Tr}(M)=c,$$

with $c\nmid P$. Existence conjecture. There exists an ordinary rank-$2$ Drinfeld $A$-module $\Phi$ over a finite field $L$ whose associated Frobenius matrix $M_F$ satisfies

$$M_F=M\in\mathbf{M}_{2}(A/\chi_{\Phi}).$$

The conjecture is proposed as a tool for proving the preceding theorem on the realization of finite $A$-modules as groups $L^{\Phi}$; the supplied text gives no resolution or further evidence for it.