The field-of-rationality lower-bound conjecture for positive-depth supercuspidals

Let $G/F$ be a connected reductive group, and let $|W|$ be the cardinality of the Weyl group of a maximal torus in $G\times\overline{F}$. Let $\mathfrak{p}$ be a prime such that $F_{\mathfrak{p}}$ has sufficiently high residue characteristic $p$, depending only on $G$. If $\pi$ is a supercuspidal representation of $G(F_{\mathfrak{p}})$ of positive depth, the field-of-rationality lower-bound conjecture asserts that

$$[\mathbb{Q}(\pi):\mathbb{Q}]\geq\frac{p-1}{|W|}.$$

This would give a uniform lower bound, growing with the residue characteristic, for the degrees of fields of rationality of positive-depth supercuspidal representations; the text presents it as a potential approach and cautions that the details have not been checked.