Conjecture on projective special linear and unitary Miyamoto groups over prime fields

Let $F$ be a prime field, let $\alpha,\beta,\gamma,\psi\in F$ be parameters, and let $\tau_a,\tau_b,\tau_c$ be the corresponding Miyamoto involutions. Define

$$T=\langle\tau_a,\tau_b,\tau_c\rangle.$$

Projective-group conjecture. There exist parameters $\alpha,\beta,\gamma,\psi\in F$ such that $T\simeq PSL(3,F)$, and a different set of parameters such that $T\simeq PSU(3,F)$. The preceding computations provide examples over the prime fields of orders $7$, $11$, and $13$ for both families, motivating the proposed existence statement for every prime field.