Gauss-sum local converse conjecture in prime characteristic

Let $n$ be prime and let $\omega$ be a character of $\mathbb F_{p^n}^{\times}$; write $S(\chi)$ for the corresponding Gauss sum. Let $\hat k=k(p^n-1)/(p-1)$. Prime-characteristic Gauss-sum conjecture. If $\omega^{-\alpha}$ and $\omega^{-\beta}$ are regular and

$$S(\omega^{-(\alpha+\hat k)})=S(\omega^{-(\beta+\hat k)})$$

for every $0\le k<p-1$, then

$$\alpha\equiv p^j\beta\pmod{p^n-1}$$

for some $j$. This is a reformulation of the preceding Gauss-sum conjecture using exponents and norm twists; the source does not give a general proof.