Pairwise coprimality conjecture for prime-indexed Lucas sequence terms

Let $(u_n)_{n\ge 0}$ be the Lucas sequence associated to integers $a$ and $b$ with $|a|\ge 3$ and $b=-1$, so that $u_0=0$, $u_1=1$, and

$$u_{n+1}=a u_n+b u_{n-1}.$$

Pairwise coprimality conjecture. For any two different primes $p$ and $q$, the terms $u_p$ and $u_q$ are relatively prime:

$$\operatorname{gcd}(u_p,u_q)=1.$$

The preceding theorem proves that $|u_n|$ is composite for every $n\ge 3$, while this conjecture asserts that prime-indexed terms nevertheless do not share any common prime divisor. If true, it would imply that no finite set of primes divides every term of the sequence.