The Drinfeld analogue of Ogg's rational cyclic-subgroup conjecture

Let $A=\mathbb{F}_q[T]$, let $F$ be its fraction field, and let $\phi$ be a rank-$2$ Drinfeld $A$-module over $F$. Define

$$C=\begin{cases}3,&q\neq 3,\\4,&q=3.\end{cases}$$

Drinfeld cyclic-subgroup conjecture. If $\mathfrak{p}$ is a prime of $A$ with $\deg\mathfrak{p}\geq C$, then there is no $F$-rational $A$-submodule of $\phi[\mathfrak{p}]$ isomorphic to $A/\mathfrak{p}$. Equivalently, $Y_0(\mathfrak{p})$ has no $F$-rational points. The conjecture is motivated by the genus of $X_0(\mathfrak{p})$ and known rational CM points. No resolution is supplied.