Conjecture on sums of a 5-potent and an n-potent in finite fields

Let $q$ be a prime power and let $n$ satisfy the normal conditions for the Restricted Problem. An element $a\in\mathbb{F}_q$ is 5-potent if it is a fifth power, and $n$-potent if it is an $n$th power. Finite-field 5-potent sum conjecture. Every element of $\mathbb{F}_q$ is a sum of a 5-potent and an $n$-potent only for the pairs

$$(q,n)\in\{(7,4),(9,3),(9,5),(5,2),(5,3),(13,5),(13,7),(17,9),(25,9),(25,13),(29,15),(41,21),(49,25),(53,27),(73,37),(81,41),(125,63)\}.$$

The conjecture extends the paper's Restricted Problem from $m=4$ to $m=5$; the stated list is supported by computations for $q\leq 10000$, while the theoretical argument for this case has not yet been undertaken.