Cohen's classification conjecture for sums of a 5-potent and an n-potent

Let $\mathbb{F}_q$ be a finite field of order $q=p^v$, where $p$ is prime and $v$ is a positive integer. For an integer $r>1$, call $a\in\mathbb{F}_q$ $r$-potent if $a^r=a$, and let $C_r$ denote the set of all $r$-potents. Under the normal conditions, every element of $\mathbb{F}_q$ is a sum of a 5-potent and an $n$-potent only for the pairs

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

Here the normal conditions are those imposed in the paper for the finite-field problem, including $n>1$ and the reduction to exponents compatible with $q-1$. This conjecture proposes a complete classification of finite fields satisfying the stated sum condition; the paper says that its results confirm the conjecture for all finite fields satisfying the condition.