Does some nontrivial Lucas sequence contain infinitely many primes?

The Mersenne numbers $M_n=2^n-1:n$ not necessarily prime are a Lucas sequence with recurrence relation $x_{n+1}=2x_n+1$.

It's an open problem how many Mersenne numbers are prime and we know neither whether $0\%$ or $100\%$ are prime (asymptotically speaking).

There are also similar sequences of repunits base $n$ with some nice maths surrounding them.

There are Lucas sequences having primes up to some point and then no more primes, such as the sequence with the relation $x_{n+1}=4x_n+1$ given by $1,5,21,85,341,\ldots$ for which it can be shown that there are no more primes beyond $5$.

We can also find sequences having no primes at all such as the sequence with the same relation but starting at $8$, given by $8,33,133,533,\ldots$ - and in fact it is true for any sequence for which $3x_0+1$ is a square that it has no primes - so we can say there are infinitely many Lucas sequences having no primes.

The obvious case to ask is whether infinitely many of the Fibonacci numbers are prime - and this is another open problem.

Is it known, or is it possible to show, that there is *some* (nontrivial) Lucas sequence (identifiable or otherwise), having infinitely many primes, or that there is none?