The G2 maximal cross-correlation conjecture for non-Germain prime fields

Let $p\neq 19$ be a prime that is not a Germain prime. Let $f_1$ and $f_2$ be $G_2$ permutations in the finite field $\mathbb{F}(p)$, and let $f'_1$ and $f'_2$ be $G_2$ permutations in the same field, with their second primitive root in common. Write $\Psi_{f_1,f_2}(u,v)$ for their cross-correlation at shifts $(u,v)$.

G2 cross-correlation conjecture.

$$\max_{(u,v)}\max_{(f_1\neq f_2)}\Psi_{f_1,f_2}(u,v)=\max_{(f'_1\neq f'_2)}\Psi_{f'_1,f'_2}(0,0).$$

The conjecture asserts that the maximal cross-correlation is attained at zero shifts in the indicated comparison. It is supported by numerical data in the source, which does not state a resolution.