Further Rigid Triples of Classes in $G_{2}$

We establish the existence of two rigid triples of conjugacy classes in the algebraic group $G_{2}$ in characteristic $5$, complementing results of the second author with Liebeck and Marion. As a corollary, the finite groups $G_{2}(5^n)$ are not $(2,4,5)$-generated, confirming a conjecture of Marion in this case.