I have two contacts in my messages box.
The first contact is an automated welcome message I received from a bot after joining a community on lemmy 4 months ago. The second contact is a conversation I initiated with a user of a piefed instance.
When I open either conversation, the top message showing on the screen is the one I received from the bot, and below that the conversation with the other user is showing up.
I'll try DMing you too and see what happens then.
Edit: Yes, my message to you has also been appended to all conversations for all contacts.