Thakur's conjecture on degrees of Wieferich primes

Let $A=\mathbb{F}_q[T]$ have characteristic $p$, and let a Wieferich prime in $A$ mean a prime satisfying the Wieferich condition discussed in the paper. Thakur's conjecture. For every $p>2$, the degree of a Wieferich prime in $A$ is divisible by $p$. This conjecture relates the degree of a Wieferich prime in the function-field setting to the characteristic; the supplied text gives no resolution.