Conjecture on the optimal bound for primitive pairs of type $(1,1)$

Let $Q_{1,1}$ be the set of prime powers $q$ for which every rational function in $R_{1,1}$ admits a primitive pair $(\alpha,f(\alpha))$ with $\alpha\in\mathbb{F}_q$, and let $C_{1,1}$ be the least threshold such that all prime powers $q>C_{1,1}$ belong to $Q_{1,1}$. For functions $f(x)=a(x+b)/(x+c)$ with $a,b,c\in\mathbb{F}_q^*$ and $b\ne c$, the conjecture is that $C_{1,1}=331$ and that the prime powers not in $Q_{1,1}$ are exactly those listed in Case 1. This proposes that the computationally identified exceptions are complete and that $331$ is the optimal bound.