If $n$ is a Poulet number then $n+2$ is not a Poulet number?

Question

Consider Fermat pseudoprimes to base $2$, also called Sarrus numbers or Poulet numbers.

Inspired by prime twins it makes sense to consider :

Conjecture :

If $n$ is a Poulet number then $n+2$ is not a Poulet number.

So there are no "Poulet twins".

Is there a way to prove this ?

Or to reduce it to other smaller or equivalent conjectures ?

Edit

Apparantly $4369$ and $4371$ are a counterexample.

But are there any more ?

What are good arguments for or against there being infinitely many ?

For prime twins we have sieve arguments, do we have that here ?

When looking at the list here it seems to be somewhat accelerating, so that is why I started to wonder about these twins

https://oeis.org/A001567/list

Answer 1

As is listed explicitly in the corresponding OEIS sequence, both 4369 and 4371 are base-2 Fermat pseudoprimes. There's no reason to think there aren't infinitely many such pairs.

In general, I would hesitate to make conjectures in number theory simply by concatenating number theory nouns and adjectives together. Conjectures that are fruitful should have some actual reasoning behind them (heuristics, algebraic structure, data, evidence...).