The thing with LGD killing other flocks isn't a case of genetics. It's a case of "pack". This is why a big part of LGD training includes proper intoduction to the livestock they are supposed to protect. Outside animals aren't part of the pack, so can be considered fair game to some dogs. The same goes for my dogs. My chickens and rabbits are safe. If the neighbors birds were to wander over here, it would be a massacre. Same for wild bunnies - they are fair game to the dogs. It goes for all dogs. Most of the cases I've heard are actually herding dogs, to be fair. Collies, GSDs, etc. Then of course all the family dogs (labs for example) that are fine at home but will wander down the street and kill every bird in the neighbor's flock. If I had. Adollar for every one of those, i'd be rich!
If you DO send this letter, be sure to keep a copy and send it certified so that you have proof. Otherwise, if things go south in the neighbor relations, he can't just say he didn't know. This is why I don't deal with these problems. I simply call the sheriff and have a deputy write up a report. The deputy talks to the neighbor, warns them of the law concerning livestock, and givess them an amount for the damages. I also have an official papertrail if I have to shoot the dog.
Even better, if I catch a different dog in the act, I have a papertail of all past problems. It also shows a pattern of certain neighbors who have one dog after another who is left to roam and cause problems. Here, you are allowed to shoot a dog if it is "reasonable" to believe it will kill or harm livestock. With a list of police reports detailing past losses, I am covered. I can prove that, if not this dog, there is a pattern and that I am reasonable in my belief that this dog running around my coop will be a problem, even if he hasn't found a way in yet.