Nonvanishing conjecture for the rank-three recurrence sequence

Let $\{g_n\}_{n\geq 0}$ be the sequence defined earlier in the paper through the rank-three Witten zeta function of type $G$. Nonvanishing conjecture for $g_n$. If $n\equiv 0,2,4,8\pmod {10}$, then

$$g_n\neq 0.$$

This conjecture would remove the word “possibly” from the stated pole description of $\omega_G(s)$, proving that the corresponding candidate poles are genuine. The source indicates that this should follow from an asymptotic analysis of the associated second-order recurrence, but does not provide that proof.