The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$ and $$L_0=2,\ L_1=1,\ \text{and}\ L_{n+1}=L_n+L_{n-1}\ (n=1,2,3,\ldots).$$
Here I report my following conjecture on primitive roots modulo primes related to Fibonacci or Lucas numbers.
Conjecture. (i) For any prime $p$, there are two Fibonacci numbers $F_k$ and $F_m$ with $F_kF_m<p$ such that $F_kF_m$ is a primitive root modulo $p$.
(ii) For any prime $p$, there are two Lucas numbers $L_k$ and $L_m$ with $L_kL_m<p$ such that $L_kL_m$ is a primitive root modulo $p$.
Clearly, part (i) of the conjecture implies that for each odd prime $p$ there exists a Fibonacci number $F_k<p$ which is a quadratic nonresidue modulo $p$. Similarly, part (ii) of the conjecture implies that for each odd prime $p$ there exists a Lucas number $L_k<p$ which is a quadratic nonresidue modulo $p$.
It seems that the first part of the above conjecture can be strengthened as follows: For any prime $p$, there is a primitive root $g < p$ modulo $p$ such that one of $$\frac g{F_2} = g,\ \ \frac g{F_3}=\frac g2,\ \ \frac g{F_4}=\frac g3$$ is a Fibonacci number. I have verified this strong version for all primes $p < 10^9$. For example, $F_4F_5=3\times5=15$ is a primitive root modulo the prime $439$. I have also verified the second part of the conjecture for all primes $p<10^9$.
QUESTION. How to prove the conjecture? Can one find a concrete counterexample to the strong version of the first part?
