Exceptional groups in the triality construction for the symmetric and alternating groups

The construction associates generators $\rho_1,\rho_2,\rho_3$ to a group $G$ by

$$G=\langle \rho_1,\rho_2,\rho_3\rangle.$$

A triality construction conjecture. Apart from two cases, up to coloring symmetries, the associated group satisfies $G\in\{\operatorname{A}_n,\operatorname{S}_n\}$. The exceptions occur at $n=7$ and $n=13$, where respectively

$$G\cong\operatorname{PSL}(2,7)\quad\text{and}\quad G\cong\operatorname{PSL}(3,3).$$

This statement records the exceptional outcomes of the construction rather than an independent existence assertion. The source presents it in a conjecture environment, but the supplied context does not establish whether it has been proved or remains open.