The bounded perfection exponent conjecture for Anderson modules

Let $E$ be an Anderson $A$-module over a field $R$, and let $b(E)$ be an exponent such that the residue-in-$\tau$ pairing factors through $R^{1/q^{b(E)}}$. Let $w(E)$ range over the weights of $E$. Bounded perfection exponent conjecture. One may choose $b(E)$ so that

$$|b(E)w(E)|<1$$

for all weights $w(E)$ of $E$.

This conjecture proposes a uniform bound on the exponent needed for the residue-in-$\tau$ pairing to descend from the perfection of $R$. It is motivated by several examples, but the source provides no resolution.