Conjecture on primitive elements and a square discriminant over finite fields

Let $p$ be an odd prime and let $e_1>1$ and $e_2>1$ be odd integers. For $i=1,2$, choose a primitive element $j_i\in\mathbb{F}_{p^{e_i}}$ and define

$$\omega_i=j_i+j_i^{-1}.$$

Primitive-element square conjecture. There exist such $j_1$ and $j_2$ for which

$$\Omega:=\omega_1^2\omega_2^2-4(\omega_1^2+\omega_2^2)$$

is a square in $\mathbb{F}_{p^{\operatorname{lcm}(e_1,e_2)}}$. This conjecture is introduced to make the construction of generators for chiral $4$-polytopes work in the unresolved cases; the paper reports extensive computational evidence but no proof.