Does writing letters stop villagers from leaving?

I have a good friend who plays the Gamecube game and a few of their favourite villagers moved out recently, so I wanted to try to help them find ways to make it more likely that their faves will stay. I overheard someone claiming that if you write letters to your villagers but don't send the letters, that the villagers won't leave until you send them. However, the post was very old, on a website I'm not a member of, and very vague, so I wondered if anyone could confirm here if it works or is a rumor?

I am almost positive that doesn't work. Only the last two villagers will move, everyone else is permanent. They could visit another town and hopefully some villagers would move and not their faves, if they manage that then their faves will be permanent. The villagers that move into the other town are random though so it's a risk.

This only works if they visit a town that isn't full. Everytime they visit someone will move. So say you have Tutu you want to keep forever but she's the last villager that moved into your town, visit a town twice and two random villagers will move out of your town. Well now you don't have max villagers and those two last spots are open. Now Tutu is permanent.