Conjecture on inverse pairs of primitive 1-normal elements

Let $t$ be a positive integer. A pair $(\alpha,\alpha^{-1})$ is a pair of primitive $1$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ when both elements are primitive and $1$-normal over $\mathbb{F}_q$. Inverse-pair conjecture. There always exists such a pair if $n=4$ and $q=4t+1$, if $n=5$ and $q=5^t$, or if $n=6$ and $q=2^t$ or $q=3^t$, with the sole genuine exceptions

$$(n,q)=(6,2),(6,4).$$

The claim is based on extensive experiments in the listed parameter families. The stated exceptions are asserted to be the only genuine failures, while no general proof is supplied in the source.