The exceptional-value conjecture for Engel trace maps

Let $\mathbb{F}_q$ be a finite field, let $a\in\mathbb{F}_q$, and let $n\in\mathbb{N}$. Let $\rho_n$ denote the trace polynomial introduced in the paper for the $n$-th Engel word.

Exceptional-value conjecture. Unless either $a=1$ and $\sqrt{2}\notin\mathbb{F}_q$, or the triple $(q,a,n)$ occurs in one of the explicitly listed exceptional cases, the map $\rho_n$ attains the value $a$.

This claim is based on computer experiments for $q<600$ and $n<50$. The source presents it as a conjectural description of the exceptional values, while the listed computational cases and the square-root obstruction remain excluded.