Kohl's primitive-root permutation sign conjecture

Let $p$ be an odd prime, let $g$ be a primitive root modulo $p$, and define $\sigma_g$ on $\mathbb Z/p\mathbb Z=\{0,1,\ldots,p-1\}$ by

$$\sigma_g(b):=g^b$$

for $b\in\{1,\ldots,p-1\}$, with $\sigma_g(0)=0$. Let $\mathcal R_p$ be the set of primitive roots modulo $p$, and regard $\sigma_g$ as a permutation of $\mathbb Z/p\mathbb Z$. Kohl's conjecture. (i) If $p\equiv1\pmod4$, then

$$\#\{g\in\mathcal R_p:\operatorname{sgn}(\sigma_g)=1\}=\#\{g\in\mathcal R_p:\operatorname{sgn}(\sigma_g)=-1\}.$$

(ii) If $p\equiv3\pmod4$, then for every $g\in\mathcal R_p$,

$$\operatorname{sgn}(\sigma_g)\equiv-\left(\frac{p-1}{2}\right)!\pmod p.$$

The problem was posed by S. Kohl in 2018. The supplied status evidence says that this conjecture was confirmed by Ladisch and Petrov.